\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 72, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/72\hfil Boundary-value problem for a nonlinear equation] {Solvability of characteristic boundary-value problems for nonlinear equations with iterated wave operator in the principal part} \author[S. Kharibegashvili, B. Midodashvili\hfil EJDE-2008/72\hfilneg] {Sergo Kharibegashvili, Bidzina Midodashvili} % in alphabetical order \address{Sergo Kharibegashvili \newline A. Razmadze Mathematical Institute \\ 1, M. Aleksidze St., Tbilisi 0193, Georgia} \email{kharibegashvili@yahoo.com} \address{Bidzina Midodashvili \newline A. Razmadze Mathematical Institute \\ 1, M. Aleksidze St., Tbilisi 0193, Georgia} \email{bidmid@hotmail.com} \thanks{Submitted February 18, 2008. Published May 15, 2008.} \subjclass[2000]{35L05, 35L35, 35L75} \keywords{Characteristic boundary-value problem; hyperbolic equations; \hfill\break\indent wave operator; power nonlinearity; nonexistence} \begin{abstract} A characteristic boundary-value problem for a hyperbolic equation with power nonlinearity and iterated wave operator in the principal part is considered in a conical domain. Depending on the exponent of nonlinearity and spatial dimensionality of the equation, the existence and uniqueness of the solution of a boundary-value problem is established. The non-solvability of this problem is also considered here. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{introduction} In the Euclidean space $ \mathbb{R}^{n+1}$ of independent variables $x_1 ,x_2 ,\dots,x_n ,t$, consider the nonlinear equation \begin{equation} L_\lambda u: = \square ^2 u = \lambda f(u) + F, \label{e1.1} \end{equation} where $\lambda $ is a given real constant, $f:\mathbb{R}\to\mathbb{R}$ is a given continuous nonlinear function, $f(0) = 0$, $F$ is a given, and $u$ is an unknown real functions, and for $n \geq 2$, \[ \square = \frac{{\partial ^2 }}{{\partial t^2 }} - \sum_{i = 1}^n {\frac{{\partial ^2 }}{{\partial x_i ^2 }}}\,. \] Let $D_T :| x | < t < T-|x|$ be a domain, which is the intersection of the light cone of future $K_O^ + :t > |x|$ with the apex in the origin $O(0,0,\dots,0)$ and light cone of past $K_A^{-} :t < T-|x|$ with apex in point $A(0,\dots,0,T),\,\, T=const>0$. For equation \eqref{e1.1} consider the boundary-value problem on determination of its solution $u(x_1 ,\dots,x_n ,t)$ in domain $D_T$ with the boundary condition \begin{equation} u \big|_{\partial D_T } = 0. \label{e1.2} \end{equation} It should be noted that for nonlinear hyperbolic equations the local or global solvability of the Cauchy problem with initial conditions for $t = 0$ and mixed problems has been studied in numerous publications; see, \cite{c1,c3,c4,g2,g3, h3,i1,j1,j2,j3,j4,k1,l2,m1,s2,s3,t1,y1,y2,z1}. Regarding the nonlinear wave equation $\square u = \lambda f(u) + F$, we have the following results: The characteristic problem in the light cone of future $K_O^ + :t> |x|$, with boundary condition $u |_{\partial K_O^ + } = g$, in the linear case with $\lambda = 0$, is well-posed and has global solvability in some appropriate function spaces; see \cite{b1,c2,c5,h1,l3}. Meanwhile, the nonlinear case, when $f(u)$ has exponential nature and $\lambda \ne 0$, has been considered in \cite{k2,k3,k4}. Assume ${\mathaccent"7017 C}^k (\overline D _T ,\partial D_T) = \{ {u \in C^k (\overline D _T ): u |_{\partial D_T } = 0} \}$, $k \geq 1$. Let $u \in {\mathaccent"7017 C}^4(\overline D _T$, $\partial D_T)$ be a classical solution of problem \eqref{e1.1}-\eqref{e1.2}. Multiplying the both parts of \eqref{e1.1} by an arbitrary function $\phi \in {\mathaccent"7017 C}^2(\overline D_T,\partial D_T)$ and integrating obtained equation by parts in domain $D_T $ we obtain \begin{equation} \int_{D_T } {\square u\square \phi \,dx\,dt} = \lambda \int_{D_T } {f(u)\phi \,dx\,dt} + \int_{D_T } {F\phi \,dx\,dt}. \label{e1.3} \end{equation} Here we used the equality $$ \int_{D_T } {\square u\square \phi \,dx\,dt} = \int_{\partial D_T } {\frac{{\partial \phi }} {{\partial N}}\square uds} - \int_{\partial D_T } {\phi \frac{\partial } {{\partial N}}\square uds} + \int_{D_T } {\phi \square ^2 u\,dx\,dt} $$ and the fact that since $\partial D_T$ is characteristic manifold, then derivative on the conormal \[ \frac{\partial }{{\partial N}} = \gamma _{n + 1} \frac{\partial }{{\partial t}} -\sum_{i = 1}^n {\gamma _i \frac{\partial } {{\partial x_i}}}, \] where $\gamma = (\gamma _1 ,\dots,\gamma _n ,\gamma _{n + 1} )$ is the unit vector of external normal relative to $\partial D_T $, is an inner differential operator on characteristic manifold $\partial D_T$ and, thus, if $v \in {\mathaccent"7017 C}^1 (\overline D _T ,\partial D_T)$, then ${\frac{{\partial v}}{{\partial N}}} |_{\partial D_T} = 0$. Let us introduce the Hilbert space ${\mathaccent"7017 W}^1_{2, \square }(D_T )$ as a completion with respect to the norm \begin{equation} \| u \|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )}^2 = \int_{D_T } {\big[ {u^2 + ({\frac{{\partial u}} {{\partial t}}})^2 + \sum_{i = 1}^n {({\frac{{\partial u}} {{\partial x_i }}})^2 + ({\square u})^2 } } \big]} \,dx\,dt \label{e1.4} \end{equation} of classical space ${\mathaccent"7017 C}^2 (\overline D _T ,\partial D_T)$. It follows from \eqref{e1.4} that if $u \in {\mathaccent"7017 W} _{2,\,\square }^1 (D_T )$, then $u \in {\mathaccent"7017 W} _{2}^1 (D_T )$ and $\square u \in L_2 (D_T )$. Here $W_2^1 (D_T )$ is the known Sobolev space \cite[p. 56]{l1}, consisting of elements from $L_2 (D_T )$, which have first order generalized derivatives in $L_2 (D_T )$, and ${\mathaccent"7017 W} _2^1 (D_T ) = \{ {u \in W_2^1 ({D_T }): u |_{\partial D_T } = 0} \}$, where equality $ u |_{\partial D_T } = 0$ should be understood in the sense of the theory of trace \cite[p. 70]{l1}. Let us assume \eqref{e1.3} as the basis of determination of generalized solution of problem \eqref{e1.1}-\eqref{e1.2}. \begin{definition} \label{def1} \rm Let $F \in L_2 (D_T )$. We call function $u \in {\mathaccent"7017 W}_{2,\square }^1 (D_T )$ a weak generalized solution of problem \eqref{e1.1}-\eqref{e1.2} if $f(u) \in L_2 (D_T )$ and for any function $\phi \in {\mathaccent"7017 W} _{2,\square }^1 (D_T )$ it is valid integral equality \eqref{e1.3}; i.e. \begin{equation} \int_{D_T } {\square u\square \phi \,dx\,dt} = \lambda \int_{D_T } {f(u)\phi \,dx\,dt} + \int_{D_T } {F\phi \,dx\,dt} \quad \forall \phi \in {\mathaccent"7017 W} _{2,\square }^1 (D_T ). \label{e1.5} \end{equation} \end{definition} It is easy to verify that if the solution $u$ of problem \eqref{e1.1}-\eqref{e1.2} in the sense of the above definition belongs to the class $C^4 (\overline D _T )$, then it will be a classical solution of this problem. \section{solvability of \eqref{e1.1}-\eqref{e1.2} with $f(u) = |u|^\alpha \mathop{\rm sgn}u$} Assume that for a positive constant $\alpha \ne 1$, the nonlinear function $f$ in \eqref{e1.1} has the form \begin{equation} f(u) = |u|^\alpha \mathop{\rm sgn} u\,. \label{e2.1} \end{equation} Then in accordance to \eqref{e2.1}, equation \eqref{e1.1} and \eqref{e1.5} take the form \begin{equation} L_\lambda u: = \square ^2 u = \lambda |u|^\alpha \mathop{\rm sgn} u + F \label{e2.2} \end{equation} and \begin{equation} \int_{D_T } {\square u\square \phi \,dx\,dt} = \lambda \int_{D_T } {\phi |u|^\alpha \mathop{\rm sgn} u\,dx\,dt} + \int_{D_T } {F\phi \,dx\,dt}, \quad \forall \phi \in {\mathaccent"7017 W} _{2,\square }^1 (D_T ). \label{e2.3} \end{equation} \begin{lemma} \label{lem1} With the norm of the space ${\mathaccent"7017 W} _{2,\square }^1 (D_T )$ given in \eqref{e1.4}, \begin{equation} \|u\|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )} \leq c\|\square u\|_{L_2 (D_T )} \quad \forall u \in {\mathaccent"7017 W} _{2,\square }^1 (D_T ) \label{e2.4} \end{equation} where $c$ is positive constant independent on $u$. \end{lemma} \begin{proof} Since the space ${\mathaccent"7017 C}^2 (\overline D _T ,\partial D_T)$ is the dense subspace of space ${\mathaccent"7017 W} _{2,\square }^1 (D_T )$ it is sufficient to prove that for all $u \in {mathaccent"7017 C}^2 (\overline D _T ,\partial D_T)$, \begin{equation} \|u\|^2_{W^1_{2,\square}(D^{+}_{T/2})} \leq c^2\|\square u\|^2_{L_2(D^{+}_{T/2})},\quad \|u\|^2_{W^1_{2,\square}(D^{-}_{T/2})} \leq c^2\|\square u\|^2_{L_2(D^{-}_{T/2})}, \label{e2.5} \end{equation} where $D^{+}_{T/2} = D_T \cap \{ t < T/2 \}$, $D^{-}_{T/2} = D_T \cap \{ t > T/2 \}$ and the norm $\|\cdot\|_{W^1_{2,\square}(D^{\pm}_{T/2})}$ is given by \eqref{e1.4} with $D^{\pm}_{T/2}$ instead of $D_T$. Let us prove the first inequality of \eqref{e2.5}, the second inequality can be proved in the same way. Assume $\Omega _\tau: = \overline D^{+}_{T/2} \cap \{ t = \tau \}$, $D^{+}_\tau = D^{+}_{T/2} \cap \{ t < \tau \}$, $S^{+}_\tau = \{ (x,t) \in \partial D^{+}_\tau :t = |x|\}$, $0 < \tau \leq T/2$ and $\gamma = (\gamma _1 ,\dots,\gamma _n ,\gamma _{n + 1})$ be the unit vector of outer normal relative to $\partial D^{+}_\tau $. For $u \in \mathop {C^2 }^0 (\overline D _T,\partial D_T)$, taking into account equalities $u|_{S^{+}_\tau } = 0$, $\Omega _\tau = \partial D^{+}_\tau \cap \{ t = \tau \} $ and $\gamma|_{\Omega _\tau } = (0,\dots,0,1)$, integrating by parts it is easy to obtain \begin{equation} \begin{aligned} \int_{D^{+}_\tau } {\frac{{\partial ^2 u}} {{\partial t^2 }}\frac{{\partial u}} {{\partial t}}\,dx\,dt} &= \frac{1}{2}\int_{D^{+}_\tau } {\frac{\partial } {{\partial t}}({\frac{{\partial u}} {{\partial t}}})^2 \,dx\,dt} = \frac{1} {2}\int_{\partial D^{+}_\tau } {({\frac{{\partial u}} {{\partial t}}})^2 \gamma _{n + 1} ds} \\ &= \frac{1} {2}\int_{\Omega _\tau } {({\frac{{\partial u}} {{\partial t}}})^2 dx} + \frac{1} {2}\int_{S^{+}_\tau } {({\frac{{\partial u}} {{\partial t}}})^2 \gamma _{n + 1} ds} ,\quad \tau \leq T/2, \end{aligned} \label{e2.6} \end{equation} \begin{equation} \begin{aligned} \int_{D^{+}_\tau } {\frac{{\partial ^2 u}} {{\partial x_i^2 }}\frac{{\partial u}} {{\partial t}}\,dx\,dt} & = \int_{\partial D^{+}_\tau } {\frac{{\partial u}} {{\partial x_i }}\frac{{\partial u}} {{\partial t}}\gamma _i ds} - \frac{1} {2}\int_{D^{+}_\tau } {\frac{\partial } {{\partial t }}({\frac{{\partial u}} {{\partial x_i }}})^2 \,dx\,dt} \\ & = \int_{\partial D^{+}_\tau } {\frac{{\partial u}} {{\partial x_i }}\frac{{\partial u}} {{\partial t}}\gamma _i ds} - \frac{1} {2}\int_{\partial D^{+}_\tau } {({\frac{{\partial u}} {{\partial x_i }}})^2 \gamma _{n + 1} ds} \\ & = \int_{\partial D^{+}_\tau } {\frac{{\partial u}}{{\partial x_i }} \frac{{\partial u}}{{\partial t}}\gamma _i ds} - \frac{1}{2}\int_{S^{+}_\tau } {({\frac{{\partial u}} {{\partial x_i }}})^2 \gamma _{n + 1} ds} - \frac{1} {2}\int_{\Omega _\tau } {({\frac{{\partial u}} {{\partial x_i }}})^2 dx} , \end{aligned}\label{e2.7} \end{equation} with $\tau \leq T/2$. It follows from \eqref{e2.6} and \eqref{e2.7} that \begin{equation} \begin{aligned} &\int_{D^{+}_\tau } {\square u\frac{{\partial u}} {{\partial t}}\,dx\,dt} \\ &= \int_{S^{+}_\tau } {\frac{1} {{2\gamma _{n + 1} }}\Big[ {\sum_{i = 1}^n {\big({\frac{{\partial u}} {{\partial x_i }}\gamma _{n + 1} - \frac{{\partial u}} {{\partial t}}\gamma _i }\big)} ^2 } } + ({\frac{{\partial u}} {{\partial t}}})^2 {\big({\gamma _{n + 1}^2 - \sum_{j = 1}^n {\gamma _j^2 } }\big)} \Big]ds \\ &\quad + \frac{1} {2}\int_{\Omega _\tau } {\big[ {({\frac{{\partial u}} {{\partial t}}})^2 + \sum_{i = 1}^n {({\frac{{\partial u}} {{\partial x_i }}})^2 } } \big]dx} ,\quad \tau \leq T. \end{aligned}\label{e2.8} \end{equation} Since $ u |_{S^{+}_\tau } = 0$ and operator $(\gamma _{n + 1} \frac{\partial }{{\partial x_i }} - \gamma _i \frac{\partial }{{\partial t}})$, $1 \leq i \leq n$, is an inner differential operator on $S^{+}_\tau $, then we have the equalities \begin{equation} {\Big({\frac{{\partial u}} {{\partial x_i }}\gamma _{n + 1} - \frac{{\partial u}} {{\partial t}}\gamma _i }\Big)} \big|_{S^{+}_\tau } = 0,\quad i = 1,\dots,n. \label{e2.9} \end{equation} Therefore, taking into account that $\gamma _{n + 1}^2 - \sum_{j = 1}^n {\gamma _j^2 } = 0$ on the characteristic manifold $S^{+}_\tau $, in view of \eqref{e2.8} and \eqref{e2.9}, we have \begin{equation} \int_{\Omega _\tau } {\big[ {({\frac{{\partial u}} {{\partial t}}})^2 + \sum_{i = 1}^n {({\frac{{\partial u}} {{\partial x_i }}})^2 } } \big]dx} = 2\int_{D^{+}_\tau } {\square u\frac{{\partial u}}{{\partial t}}\,dx\,dt} , \quad \tau \leq T/2. \label{e2.10} \end{equation} Assuming $w(\delta ) = \int_{\Omega _\delta } {[ {({\frac{{\partial u}} {{\partial t}}})^2 + \sum_{i = 1}^n {({\frac{{\partial u}} {{\partial x_i }}})^2 } }]dx}$, and using inequality $2\,\square u\frac{{\partial u}}{{\partial t}} \leq \varepsilon ({\frac{{\partial u}}{{\partial t}}})^2 + \frac{1}{\varepsilon }|\square u|^2$, which is valid for any positive $\varepsilon$, from \eqref{e2.10} we obtain \begin{equation} w(\delta ) \leq \varepsilon \int_0^\delta {w(\sigma )d\sigma } + \frac{1} {\varepsilon }\|\square \|_{L_2 (D^{+}_\delta )}^2 ,\quad 0 < \delta \leq T/2. \label{e2.11} \end{equation} From \eqref{e2.11}, taking into account that value $\|\square \|_{L_2 (D^{+}_\delta )}^2$ as a function of $\delta$ is non-decreasing, in view of Gronwall's lemma \cite[p. 13]{h2} it follows that \[ w(\delta ) \leq \frac{1} {\varepsilon }\|\square \|_{L_2 (D^{+}_\delta )}^2 \exp \delta \varepsilon. \] Hence, taking into account the fact that $\inf_{\varepsilon > 0} \frac{1} {\varepsilon }\exp \delta \varepsilon = e\delta$ and it is reached at $\varepsilon = \frac{1}{\delta }$, we obtain \[ w(\delta ) \leq e\delta \|\square \|_{L_2 (D^{+}_\delta )}^2 ,\,\,\,0 < \delta \leq T/2. \label{e2.12} \] From \eqref{e2.12}, in turn, it follows that \begin{equation} \int_{D^{+}_{T/2} } {[ {({\frac{{\partial u}} {{\partial t}}})^2 + \sum_{i = 1}^n {({\frac{{\partial u}} {{\partial x_i }}})^2 } }]\,dx\,dt} = \int_0^{T/2} {w(\delta )d\delta } \leq \frac{e}{8}T^2 \|\square u\|_{L_2 (D^{+}_{T/2} )}^2. \label{e2.13} \end{equation} Using the equalities $ u |_{S_{T/2} } = 0$ and $u(x,t) = \int_{|x|}^t {\frac{{\partial u(x,t)}}{{\partial t}}d\tau }$, $(x,t) \in \overline D^{+}_{T/2}$, which are valid for any function $u \in \mathop {C^2 }^0 (\overline D _T ,\partial D_T)$, by standard reasoning \cite[p. 63]{l1} we easily obtain \begin{equation} \int_{D^{+}_{T/2}} {u^2 (x,t)\,dx\,dt} \leq \frac {1}{4}T^2 \int_{D^{+}_{T/2}} {({\frac{{\partial u}}{{\partial t}}})^2 \,dx\,dt}. \label{e2.14} \end{equation} By virtue of \eqref{e2.13} and \eqref{e2.14}, we have \begin{align*} \|u\|_{{\mathaccent"7017 W} _{2,\square }^1 (D^{+}_{T/2} )}^2 & = \int_{D^{+}_{T/2}} \big[ {u^2 + ({\frac{{\partial u}}{{\partial t}}})^2 + \sum_{i = 1}^n {({\frac{{\partial u}}{{\partial x_i }}})^2 + ({\square u})^2 } } \big]\,dx\,dt\\ & \leq \big({1 + \frac{e}{8} T^2 + \frac{e} {32} T^4 }\big)\|\square \|_{L_2 ({D^{+}_{T/2}})}^2 , \end{align*} whence it follows the first inequality of \eqref{e2.5} with constant $c^2 = 1 + \frac{e}{8} T^2 + \frac{e}{32} T^4 $. The proof is complete. \end{proof} \begin{lemma} \label{lem2} Assume $F \in L_2 ({D_T })$, $0 < \alpha < 1$, and in the case when $\alpha > 1$ additionally require that $\lambda < 0$. Then for a weak generalized solution $u \in {\mathaccent"7017 W} _{2,\square }^1 (D_T )$ of \eqref{e1.1}-\eqref{e1.2} in the case with nonlinearity of form \eqref{e2.1}; i.e., problem \eqref{e2.2}-\eqref{e1.2} in the sense of integral equality \eqref{e2.3} with $|u|^\alpha \in L_2 ({D_T })$, it is valid a priori estimate \begin{equation} \|u\|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )} \leq c_1 \|F\|_{L_2 ({D_T })} + c_2 \label{e2.15} \end{equation} with non-negative constants $c_i ({T,\alpha ,\lambda })$, $i = 1,2$, which do not depend on $u, F$ and $c_1 > 0$. \end{lemma} \begin{proof} First let $\alpha > 1$ and $\lambda < 0$. Assuming in \eqref{e2.3} that $\phi = u \in {\mathaccent"7017 W} _{2,\square }^1 (D_T )$ and taking into account \eqref{e1.4}, for any $\varepsilon > 0$ we have \begin{equation} \begin{aligned} \|\square u\|_{L_2 ({D_T })}^2 &= \int_{D_T } {({\square u})^2 \,dx\,dt}\\ &= \lambda \int_{D_T } {|u|^{\alpha + 1} \,dx\,dt} + \int_{D_T } {Fu\,dx\,dt} \\ &\leq \int_{D_T } {Fu\,dx\,dt} \\ &\leq \frac{1}{{4\varepsilon }}\int_{D_T } {F^2 \,dx\,dt} + \varepsilon \|u\|_{L_2 ({D_T })}^2 \\ &\leq \frac{1}{{4\varepsilon }}\|F\|_{L_2 ({D_T })}^2 + \varepsilon \|u\|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )}^2. \end{aligned} \label{e2.16} \end{equation} Due to \eqref{e2.4} and the above inequality we have $$ \|u\|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )}^2 \leq c^2 \|\square u\|_{L_2 ({D_T })}^2 \leq \frac{{c^2 }}{{4\varepsilon }}\|F\|_{L_2 ({D_T })}^2 + c^2 \varepsilon \|u\|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )}^2, $$ from which for $\varepsilon = \frac{1}{{2c^2 }} < \frac{1}{{c^2 }}$, we obtain $$ \|u\|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )}^2 \leq \frac{{c^2 }} {{4\varepsilon ({1 - \varepsilon c^2 })}}\|F\|_{L_2 ({D_T })}^2 = c^4 \|F\|_{L_2 ({D_T })}^2. %\label{e2.17} $$ From this inequality in the case $\alpha > 1$ and $\lambda < 0$ follows inequality \eqref{e2.15} with $c_1 = c^2$ and $c_2 = 0$. Now let $0 < \alpha < 1$. Using the known inequality $$ ab \leq \frac{{\varepsilon a^p }}{p} + \frac{{b^q }} {{q\varepsilon ^{q - 1} }} $$ with parameter $\varepsilon > 0$ for $a = |u|^{\alpha + 1}$, $b = 1$, $p = \frac{2}{{\alpha + 1}} > 1$, $q = \frac{2}{{1 - \alpha }}$, $\frac{1}{p} + \frac{1}{q} = 1$, in the same way as for inequality \eqref{e2.16}, we have \begin{equation} \begin{aligned} &\|\square u\|_{L_2 ({D_T })}^2\\ &= \int_{D_T } {({\square u})^2 \,dx\,dt} \\ &= \lambda \int_{D_T } {|u|^{\alpha + 1} \,dx\,dt} + \int_{D_T } {Fu\,dx\,dt} \\ & \leq |\lambda |\int_{D_T } {\big[ {\varepsilon \frac{{1 + \alpha }} {2}|u|^2 + \frac{{1 - \alpha }}{{2\varepsilon ^{q - 1} }}} \big]\,dx\,dt} + \frac{1} {{4\varepsilon }}\int_{D_T } {F^2 \,dx\,dt} + \varepsilon \int_{D_T } {u^2 \,dx\,dt} \\ &= \frac{1}{{4\varepsilon }}\|F\|_{L_2 ({D_T })}^2 + \varepsilon ({|\lambda |\frac{{1 + \alpha }}{2} + 1}) \|u\|_{L_2 ({D_T })}^2 + |\lambda |\frac{{1 - \alpha }}{{2\varepsilon ^{q - 1} }}\mathop{\rm meas}D_T. \end{aligned}\label{e2.18} \end{equation} In view of \eqref{e1.4} and \eqref{e2.4} it follows from \eqref{e2.18} that \begin{align*} &\|u\|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )}^2\\ & \leq c^2 \|\square u\|_{L_2 ({D_T })}^2 \\ & \leq \frac{{c^2 }}{{4\varepsilon }}\|F\|_{L_2 ({D_T })}^2 + \varepsilon c^2 ({|\lambda |\frac{{1 + \alpha }} {2} + 1})\|u\|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )}^2 + c^2 |\lambda |\frac{{1 - \alpha }}{{2\varepsilon ^{q - 1} }}\mathop{\rm meas}D_T , \end{align*} where $q = \frac{2}{{1 - \alpha }}$; whence for $\varepsilon = \frac{1}{2}c^{ - 2} ({|\lambda |\frac{{1 + \alpha }} {2} + 1})^{ - 1} $, \begin{equation} \begin{aligned} &\|u\|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )}^2 \\ &\leq \big[ {1 - \varepsilon c^2 \big({|\lambda | \frac{{1 + \alpha }}{2} + 1}\big)} \big]^{ - 1} \Big({\frac{{c^2 }} {{4\varepsilon }}\|F\|_{L_2 ({D_T })}^2 + c^2 |\lambda |\frac{{1 - \alpha }} {{2\varepsilon ^{q - 1} }}\mathop{\rm meas} \mathop{\rm meas}D_T }\Big) \\ &= c^4 \big({|\lambda |\frac{{1 + \alpha }}{2} + 1}\big) \|F\|_{L_2 ({D_T })}^2 + 2c^2 |\lambda | \frac{{1 - \alpha }}{{2\varepsilon ^{q - 1} }}\mathop{\rm meas} D_T. \end{aligned}\label{e2.19} \end{equation} From \eqref{e2.19}, in the case when $0 < \alpha < 1$, follows inequality \eqref{e2.15} with $c_1 = c^2 ({|\lambda |\frac{{1 + \alpha }}{2} + 1})^{1/2} $ and $c_2 = c({2|\lambda |\frac{{1 - \alpha }}{{2\varepsilon ^{q - 1} }} \mathop{\rm meas}D_T })^{1/2}$, where $q = \frac{1}{{1 - \alpha }}$. The proof is complete. \end{proof} \begin{remark} \label{rmk1}\rm From the proof of Lemma \ref{lem2} it follows that in estimate \eqref{e2.15} the constants $c_1$ and $c_2$ are equal: \begin{gather} \alpha > 1,\quad \lambda < 0:\quad c_1 = c^2 ,\quad c_2 = 0; \label{e2.20} \\ 0 < \alpha < 1,\quad - \infty < \lambda < + \infty : \notag\\ c_1 = c^2 ({|\lambda |\frac{{1 + \alpha }}{2} + 1})^{1/2}, \quad c_2 = c({2|\lambda |\frac{{1 - \alpha }}{{2\varepsilon ^{q - 1} }} \mathop{\rm meas}D_T })^{\frac{1} {2}}, \label{e2.21} \end{gather} where constant $c = ({1 + \frac{e}{2}T^2 + \frac{e}{2}T^4 })^{1/2}$ is taken from estimate \eqref{e2.4}, and $q = \frac{2}{{1 - \alpha }}$. \end{remark} \begin{remark} \label{rmk2} \rm Below, we will consider a linear problem appropriate for \eqref{e1.1}-\eqref{e1.2}; i.e., when $\lambda = 0$. In this case for $F \in L_2 ({D_T })$ it is analogously introduced a concept of the weak generalized solution $u \in {\mathaccent"7017 W} _{2,\square }^1 (D_T )$ of this problem, when \begin{equation} ({u,\phi })_\square : = \int_{D_T } {\square u\square \phi \,dx\,dt} = \int_{D_T } {F\phi \,dx\,dt} \quad \forall \phi \in {\mathaccent"7017 W} _{2,\square }^1 (D_T ). \label{e2.22} \end{equation} \end{remark} \begin{remark} \label{rmk3} \rm In view of \eqref{e1.4} and \eqref{e2.4}, taking into account that \begin{align*} | {({\square u,\square \phi })_{L_2 ({D_T })} } | &= \big| {\int_{D_T } {\square u\square \phi \,dx\,dt} } \big| \\ &\leq \| {\square u} \|_{L_2 ({D_T })} \| {\square \phi } \|_{L_2 ({D_T })}\\ &\leq \| {\square u} \|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )} \| {\square \phi } \|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )}, \end{align*} the bilinear form $$ ({u,\phi })_\square : = \int_{D_T } {\square u\square \phi \,dx\,dt} $$ in \eqref{e2.22} can be considered as a scalar product in the Hilbert space ${\mathaccent"7017 W} _{2,\square }^1 (D_T )$. Therefore, since for $F \in L_2 ({D_T })$ $$ \big| {\int_{D_T } {F\phi \,dx\,dt} } \big| \leq \|F\|_{L_2 ({D_T })} \|\phi \|_{L_2 ({D_T })} \leq \|F\|_{L_2 ({D_T })} \|\phi \|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )}, $$ then due to the Riesz theorem \cite[p. 83]{g1} there is unique function $u$ in the space ${\mathaccent"7017 W} _{2,\square }^1 (D_T )$, which satisfies equality \eqref{e2.22} for any $\phi \in {\mathaccent"7017 W} ^{1}_{2,\square } (D_T )$ and for the norm of which it is valid estimate \begin{equation} \|u\|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )} \leq \|F\|_{L_2 ({D_T })}. \label{e2.23} \end{equation} Thus, introducing notation $u = L_0^{ - 1} F$, we obtain that to the linear problem appropriate to \eqref{e1.1}-\eqref{e1.2}; i.e., when $\lambda = 0$, corresponds the linear, bounded operator $$ L_0^{ - 1} :L_2 ({D_T }) \to {\mathaccent"7017 W} _{2,\square }^1 (D_T ), $$ for the norm of which, by \eqref{e2.23}, it is valid the estimate \begin{equation} \| {L_0^{ - 1} } \|_{L_2 ({D_T }) \to {\mathaccent"7017 W} _{2,\square }^1 (D_T )} \leq \| F \|_{L_2 ({D_T })}. \label{e2.24} \end{equation} Taking into account Definition \ref{def1} and Remark \ref{rmk3}, Equality \eqref{e2.3} and Problem \eqref{e2.2}-\eqref{e1.2} can be rewritten in the equivalent form \begin{equation} u = L_0^{ - 1} [ {\lambda |u|^\alpha \mathop{\rm sgn} u + F}] \label{e2.25} \end{equation} in the Hilbert space ${\mathaccent"7017 W} _{2,\square }^1 (D_T )$. \end{remark} \begin{remark} \label{rmk4} \rm The embedding operator $I:{\mathaccent"7017 W} _{2}^1 (D_T ) \to L_q (D_T )$ is a linear continuous compact operator for $1 < q < \frac{{2(n + 1)}}{{n - 1}}$, when $n \geq 2$ \cite[p. 81]{l1}. At the same time the operator of Nemytskii $N:L_q (D_T ) \to L_2 (D_T )$, which acts according to the formula $Nu = \lambda |u|^\alpha \mathop{\rm sgn} u$, $\alpha > 1$, is continuous and bounded for $q \geq 2\alpha $ \cite[p. 349]{k5}, \cite[pp. 66, 67]{k6}. Thus, if $1 < \alpha < \frac{{n + 1}}{{n - 1}}$, then there exists such number $q$, that $1 < 2\alpha \leq q < \frac{{2({n + 1})}}{{n - 1}}$ and hence the operator \begin{equation} N_1 = NI:{\mathaccent"7017 W} _2^1 (D_T ) \to L_2 (D_T ) \label{e2.26} \end{equation} is continuous and compact operator. In this case since $u \in {\mathaccent"7017 W} _2^1 (D_T )$ then it is clear that $f(u) = |u|^\alpha \mathop{\rm sgn} u \in L_2 (D_T )$. Further, since in view of \eqref{e1.4} the space ${\mathaccent"7017 W} _{2,\square }^1 (D_T )$ is continuously embedded in the space ${\mathaccent"7017 W} _2^1 (D_T )$, then taking into account \eqref{e2.26} the operator \begin{equation} N_2 = NII_1 :{\mathaccent"7017 W} _{2,\square }^1 (D_T ) \to L_2 (D_T ), \label{e2.27} \end{equation} where $I_1 :{\mathaccent"7017 W} _{2,\square }^1 (D_T ) \to {\mathaccent"7017 W} _2^1 (D_T )$ is the embedding operator, continuous and compact for $1 < \alpha < \frac{{n + 1}}{{n - 1}}$. For $0 < \alpha < 1$ operator \eqref{e2.27} is also continuous and compact, since according to the Rellich theorem \cite[p. 64]{l1} the space ${\mathaccent"7017 W} _2^1 (D_T )$ is continuously and compactly embedded into $L_2 (D_T )$, and the space $L_2 (D_T )$, in turn, is continuously embedded into $L_p (D_T)$ for $p < 2$. \end{remark} Let us rewrite equation \eqref{e2.25} in the form \begin{equation} u = Au: = L_0^{ - 1} ({N_2 u + F}), \label{e2.28} \end{equation} where the operator $N_2 :{\mathaccent"7017 W} _{2,\square }^1 (D_T ) \to L_2 (D_T )$, for $0 < \alpha < \frac{{n + 1}}{{n - 1}}$, $\alpha \ne 1$, is continuous and compact in view of the Remark \ref{rmk4}. Then taking into account \eqref{e2.24} operator $A:{\mathaccent"7017 W} _{2,\square }^1 (D_T ) \to {\mathaccent"7017 W} _{2,\square }^1 (D_T )$ in \eqref{e2.28} is also continuous and compact. At the same time according to a priori estimate \eqref{e2.15} of the Lemma \ref{lem2}, in which the constants $c_1$ and $c_2$ are given by equalities \eqref{e2.20} and \eqref{e2.21}, for any parameter $\tau \in [0,1]$ and for any solution $u \in {\mathaccent"7017 W} _{2,\square }^1 (D_T )$ of equation $u = \tau Au$ with this parameter it is valid a priori estimation \eqref{e2.15} with constants $c_1>0$ and $c_2 \geq 0$, not depending on $u,\,\tau $ and $F$. Therefore, according to the Lere-Schauder theorem \cite[p. 375]{t2} equation \eqref{e2.28}, and consequently problem \eqref{e2.2}-\eqref{e1.2} has at least one weak generalized solution $u$ in the space ${\mathaccent"7017 W} _{2,\square }^1 (D_T )$. This is summarized in the following result. \begin{theorem} \label{thm1} Let $0<\alpha<\frac {n+1}{n-1}$, $\alpha \neq 1$, $\lambda \neq 0 $ and $\lambda <0$ when $\alpha>1$. Then for any $F \in L_2 ({D_T })$ problem \eqref{e2.2}-\eqref{e1.2} has at least one weak generalized solution $u \in {\mathaccent"7017 W} _{2,\square }^1 (D_T )$. \end{theorem} \section{Uniqueness of solution for \eqref{e1.1}-\eqref{e1.2} when $f(u) = |u|^\alpha \mathop{\rm sgn} u$} Let $F \in L_2 ({D_T })$, and $u_1$, $u_2$ be two weak generalized solutions of \eqref{e2.2}-\eqref{e1.2} in the space ${\mathaccent"7017 W} _{2,\square }^1 (D_T )$. According to \eqref{e2.3}, \begin{equation} \int_{D_T } {\square u_i \square \phi \,dx\,dt} = \lambda \int_{D_T } {\phi |u_i |^\alpha \mathop{\rm sgn} u_i \,dx\,dt} + \int_{D_T } {F\phi \,dx\,dt} \quad \forall \phi \in {\mathaccent"7017 W} _{2,\square }^1 (D_T ) \label{e3.1} \end{equation} and $|u_i |^\alpha \in L_2 ({D_T })$, $i = 1,2$. For the difference $v = u_2 - u_1 $ from \eqref{e3.1} it follows that \begin{equation} \int_{D_T } {\square v\square \phi \,dx\,dt} = \lambda \int_{D_T } {\phi ({|u_2 |^\alpha \mathop{\rm sgn} u_2 - |u_1 |^\alpha \mathop{\rm sgn} u_1 })\,dx\,dt} \quad \forall \phi \in {\mathaccent"7017 W} _{2,\square }^1 (D_T ). \label{e3.2} \end{equation} Assuming $\phi = v \in {\mathaccent"7017 W} _{2,\square }^1 (D_T )$ in the above equality, we obtain \begin{equation} \int_{D_T } {({\square v})^2 \,dx\,dt} = \lambda \int_{D_T } {({|u_2 |^\alpha \mathop{\rm sgn} u_2 - |u_1 |^\alpha \mathop{\rm sgn} u_1 })({u_2 - u_1 })\,dx\,dt}. \label{e3.3} \end{equation} Let us note that for the finite values of $u_1$ and $u_2$ with $\alpha > 0$ it is valid the inequality \begin{equation} ({|u_2 |^\alpha \mathop{\rm sgn} u_2 - |u_1 |^\alpha \mathop{\rm sgn} u_1 })({u_2 - u_1 }) \geq 0. \label{e3.4} \end{equation} From \eqref{e3.3} and inequality \eqref{e3.4}, which is true for almost all points $(x,t) \in D_T $ with $u_i \in {\mathaccent"7017 W} _{2,\square }^1 (D_T )$, $i = 1, 2$, in the case when $\alpha > 0$ and $\lambda < 0$ it follows that $$ \int_{D_T } {({\square v})^2 \,dx\,dt} \leq 0, $$ whence, due to \eqref{e2.4}, we obtain $v=0$; i.e. $u_1=u_2$. This result is summarized in the next theorem. \begin{theorem} \label{thm2} Let $\alpha > 0$, $\alpha \ne 1$ and $\lambda < 0$. Then for any $F \in L_2 ({D_T })$, Problem \eqref{e2.2}-\eqref{e1.2} cannot have more than one generalized solution in ${\mathaccent"7017 W} _{2,\square }^1 (D_T )$. \end{theorem} The following result follows from Theorems \ref{thm1} and \ref{thm2}. \begin{theorem} \label{thm3} Let $0 < \alpha < \frac{{n + 1}}{{n - 1}}$, $\alpha \ne 1$ and $\lambda < 0$. Then for any $F \in L_2 ({D_T })$, Problem \eqref{e2.2}-\eqref{e1.2} has an unique weak generalized solution $u \in {\mathaccent"7017 W} _{2,\square }^1 (D_T )$. \end{theorem} \section {Non-solvability of \eqref{e1.1}-\eqref{e1.2} when $f(u) = |u|^\alpha $} Now assume that in \eqref{e1.1}, and therefore in \eqref{e1.3}, that $f(u) = |u|^\alpha$, $\alpha > 1$. \begin{theorem} \label{thm4} Let $F^0 \in L_2 ({D_T }),\| {F^0 } \|_{L_2 ({D_T })} \ne 0$, $F^0 \geq 0$, and $F = \mu F^0$, $\mu$ is a positive constant. Then when $f(u) = |u|^\alpha$ with $\alpha > 1$ and $\lambda > 0$, there exists a number $\mu _0 = \mu _0 ({F^0 ,\lambda ,\alpha }) > 0$ suh that for $\mu > \mu _0 $, problem \eqref{e1.1}-\eqref{e1.2} can not have a weak generalized solution in the space ${\mathaccent"7017 W} _{2,\square }^1 (D_T )$. \end{theorem} \begin{proof} Let us assume that there is a solution $u \in {\mathaccent"7017 W} _{2,\square }^1 (D_T )$ of problem \eqref{e1.1}-\eqref{e1.2} exists for any fixed $\mu > 0$. Then \eqref{e1.5} takes the form \begin{equation} \int_{D_T } {\square u\square \phi \,dx\,dt} = \lambda \int_{D_T } {|u|^\alpha \phi \,dx\,dt} + \mu \int_{D_T } {F^0 \phi \,dx\,dt} \quad \forall \phi \in {\mathaccent"7017 W} _{2,\square }^1 (D_T ). \label{e4.1} \end{equation} It is easy to verify that \begin{equation} \int_{D_T } {\square u\square \phi \,dx\,dt} = \int_{D_T } {u\square ^2 \phi \,dx\,dt} \quad \forall \phi \in {\mathaccent"7017 C}^4 (\overline D _T ,\partial D_T), \label{e4.2} \end{equation} where ${\mathaccent"7017 C}^4 (\overline D _T ,\partial D_T ) = \{ u \in C^4 ( {\overline D _T } ): u |_{\partial D_T } = 0 \} \subset {\mathaccent"7017 W} _{2,\square }^1 (D_T )$. Indeed, since $u \in {\mathaccent"7017 W} _{2,\square }^1 (D_T )$, and the space ${\mathaccent"7017 C}^2 (\overline D _T ,\partial D_T )$ is dense in ${\mathaccent"7017 W} _{2,\square }^1 (D_T )$, there exists such sequence $u_k \in {\mathaccent"7017 C}^2 (\overline D _k ,\partial D_k )$ that \begin{equation} \lim_{k \to \infty } \| {u_k - u} \|_{{\mathaccent"7017 W} _{2,\square }^1 (D_T )} = 0. \label{e4.3} \end{equation} Taking into account that \begin{equation} \int_{D_T } {\square u_k \square \phi \,dx\,dt} = \int_{\partial D_T } {\frac{{\partial u_k }} {{\partial N}}\square \phi ds} - \int_{\partial D_T } {u_k \frac{\partial } {{\partial N}}\square \phi ds} + \int_{D_T } {u_k \square ^2 \phi \,dx\,dt}, \label{e4.4} \end{equation} where the derivative on the conormal $\frac{\partial }{{\partial N}} = \gamma _{n + 1} \frac{\partial } {{\partial t}} - \sum_{i = 1}^n {\gamma _i \frac{\partial }{{\partial x_i }}}$ is an inner differential operator on characteristic manifold $\partial D_T $, and, therefore $ {\frac{{\partial u_k }}{{\partial N}}} |_{\partial D_T} = 0$, since ${u_k } |_{\partial D_T} = 0$, then from \eqref{e4.4} we obtain \begin{equation} \int_{D_T } {\square u_k \square \phi \,dx\,dt} = \int_{D_T } {u_k \square ^2 \phi \,dx\,dt}, \label{e4.5} \end{equation} where $\gamma=(\gamma_1,\dots,\gamma_n,\gamma_{n+1})$ is the unit vector of outer normal relative to $\partial D_T$. Passing in \eqref{e4.5} to the limit with $k \to \infty $, in view of \eqref{e1.4} and \eqref{e4.3}, we obtain \eqref{e4.2}. Taking into account \eqref{e4.2} let us rewrite equality \eqref{e4.1} in the form \begin{equation} \lambda \int_{D_T } {|u|^\alpha \phi \,dx\,dt} = \int_{D_T } {u\square ^2 \phi \,dx\,dt} - \mu \int_{D_T } {F^0 \phi \,dx\,dt} \quad \forall \phi \in {\mathaccent"7017 C}^4 (\overline D _T ,\partial D_T). \label{e4.6} \end{equation} Below we use the method of test functions \cite[p. 10-12]{k5}. Let us select such a test function $\phi \in {\mathaccent"7017 C}^4 (\overline D _T ,\partial D_T)$, that $\phi |_{D_T } > 0$. If in Young's inequality with parameter $\varepsilon > 0$ $$ ab \leq \frac{\varepsilon }{\alpha }a^\alpha + \frac{1}{{\alpha '\varepsilon ^{\alpha ' - 1} }}b^{\alpha '},\quad a,b \geq 0,\; \alpha ' = \frac{\alpha }{{\alpha - 1}} $$ we take $a = |u|\phi ^{1/\alpha}$, $b = \frac{{|\square ^2 \phi |}}{{\phi ^{\frac{1}{\alpha }} }}$, then due to the fact that $\frac{{\alpha '}}{\alpha } = \alpha ' - 1$, we have \begin{equation} |u\square ^2 \phi | = |u|\phi ^{\frac{1}{\alpha }} \frac{{|\square ^2 \phi |}} {{\phi ^{\frac{1}{\alpha }} }} \leq \frac{\varepsilon }{\alpha }|u|^\alpha \phi + \frac{1} {{\alpha '\varepsilon ^{\alpha ' - 1} }}\frac{{|\square ^2 \phi |^{\alpha '} }} {{\phi ^{\alpha ' - 1} }}. \label{e4.7} \end{equation} By \eqref{e4.7} and \eqref{e4.6} we have the inequality \[ ({\lambda - \frac{\varepsilon } {\alpha }})\int_{D_T } {|u|^\alpha \phi \,dx\,dt} \leq \frac{1} {{\alpha '\varepsilon ^{\alpha ' - 1} }}\int_{D_T } {\frac{{|\square ^2 \phi |^{\alpha '} }} {{\phi ^{\alpha ' - 1} }}\,dx\,dt} - \mu \int_{D_T } {F^0 \phi \,dx\,dt}; \] whence for $\varepsilon < \lambda \alpha $ we obtain \begin{equation} \int_{D_T } {|u|^\alpha \phi \,dx\,dt} \leq \frac{\alpha } {{({\lambda \alpha - \varepsilon })\alpha '\varepsilon ^{\alpha ' - 1} }} \int_{D_T } {\frac{{|\square ^2 \phi |^{\alpha '} }}{{\phi ^{\alpha ' - 1} }}\,dx\,dt} - \frac{{\alpha \mu }} {{\lambda \alpha - \varepsilon }}\int_{D_T } {F^0 \phi \,dx\,dt}. \label{e4.8} \end{equation} Taking into account the equalities $\alpha ' = \frac{\alpha }{{\alpha - 1}}$, $\alpha = \frac{{\alpha '}} {{\alpha ' - 1}}$, and \[ \min_{0 < \varepsilon < \lambda \alpha } \frac{\alpha } {{({\lambda \alpha - \varepsilon })\alpha '\varepsilon ^{\alpha ' - 1} }} = \frac{1} {{\lambda ^{\alpha '} }}, \] which is reached at $\varepsilon = \lambda $, it follows from \eqref{e4.8} that \[ \int_{D_T } {|u|^\alpha \phi \,dx\,dt} \leq \frac{1} {{\lambda ^{\alpha '} }}\int_{D_T } {\frac{{|\square ^2 \phi |^{\alpha '} }} {{\phi ^{\alpha ' - 1} }}\,dx\,dt} - \frac{{\alpha '\mu }} {\lambda }\int_{D_T } {F^0 \phi \,dx\,dt}. \label{e4.9} \] Let us note that is not difficult to the existence of test function $\phi $, such that \begin{equation} \phi \in {\mathaccent"7017 C}^4 (\overline D _T ,\partial D_T),\quad \phi|_{D_T } > 0,\quad \kappa = \int_{D_T } {\frac{{|\square ^2 \phi |^{\alpha '} }} {{\phi ^{\alpha ' - 1} }}\,dx\,dt} < + \infty \,. \label{e4.10} \end{equation} Indeed, it is easy to verify that the function $$ \phi (x,t) = \big[ {({t^2 - |x|^2 })({({T - t})^2 - |x|^2 })} \big]^m $$ for sufficiently large positive $m$ satisfies conditions \eqref{e4.10}. According to the conditions in this theorem, $F^0 \in L_2 ({D_T })$, $\| {F^0 } \|_{L_2 ({D_T })} \ne 0$, $F^0 \geq 0$, and $\mathop{\rm meas}D_T < + \infty $. Then due to the fact that $\phi |_{D_T } > 0$ we have \begin{equation} 0 < \kappa _1 = \int_{D_T } {F^0 \phi \,dx\,dt < + \infty }. \label{e4.11} \end{equation} Let us denote by $g(\mu)$ the right side of inequality \eqref{e4.9}, which is a linear function with respect to $\mu $, then in view of \eqref{e4.10} and \eqref{e4.11} we have \begin{equation} g(\mu ) < 0 \text{ for }\mu > \mu _0 \quad \text{and}\quad g(\mu ) > 0 \text{ for }\mu < \mu _0, \label{e4.12} \end{equation} where $$ g(\mu ) = \frac{{\kappa _0 }} {{\lambda ^{\alpha '} }} - \frac{{\alpha '\mu }}{\lambda }\kappa _1 ,\quad \mu _0 = \frac{\lambda }{{\alpha '\lambda ^{\alpha '} }}\frac{{\kappa _0 }} {{\kappa _1 }} > 0. $$ According to \eqref{e4.12} with $\mu > \mu _0 $ the right side of inequality \eqref{e4.9} is negative, while the left side is non-negative. This contradiction completes the proof. \end{proof} \begin{thebibliography}{00} \bibitem{b1} A. V. Bitsadze; \emph{Some Classes of Partial Differential Equations}. Izdat. "Nauka", Moscow, 1981. \bibitem{c1} M. M. Cavalcanti, V. N. D. Cavalcanti, I. Lasiecka; \emph{Well posedness and optimal decay rates for the wave equation with nonlinear boundary damping - source interaction}. J. Differential Equations \textbf{236} (2007), 407-459. \bibitem{c2} F. Cagnac; \emph{Probl\`{e}me de Cauchy sur un cono\"{\i}de caract\'{e}ristique}. Ann. Mat. Pura Appl. (4) \textbf{104} (1975), 355-393. \bibitem{c3} R. C. Char\~{a}o and R. Ikehata; \emph{Decay of solutions for a semilinear system of elastic waves in an exterior domain with damping near infinity}. Nonlinear Analysis \textbf{67} (2007), 398-429. \bibitem{c4} J. W. Cholewa and T. Dlotco; \emph{Strongly damped wave equation in uniform spaces}. Nonlinear Analysis \textbf{64} (2006), 174-187. \bibitem{c5} R. Courant; \emph{Partial Differential Equations}. Izdat. "Mir", Moscow, 1964. \bibitem{g1} D. Gilbarg and N. Trudinger; \emph{Elliptic Partial Differential Equations of Second Order}. Izdat. ``Nauka'', Moscow, 1989. \bibitem{g2} V. Georgiev, H. Lindblad, and C.D. Sogge; \emph{Weighted Strichartz estimates and global existence for semilinear wave equations}. Amer. J. Math. \textbf{119}(1997), no.6, 1291-1319. \bibitem{g3} J. Ginibre, A. Soffer, and G. Velo; \emph{The global Cauchy problem for the critical nonlinear wave equation}. J Funct. Anal. \textbf{110}(1992), no.1, 96-130. \bibitem{h1} J. Hadamard; \emph{Le probl\`{e}me de Cauchy et les \'{e}quations aux d\'{e}riv\'{e}es partielles lin\'{e}aires hyperboliques}. Hermann, Paris, 1932. \bibitem{h2} D. Henry, Geometric Theory of Semilinear Parabolic Equations. Izdat. "Mir", Moscow, 1985. \bibitem{h3} L. H\"{o}rmander; \emph{Lectures on Nonlinear Hyperbolic Differential Equations}. Math\'{e}matics \& Applications (Berlin), Vol.26, Springer-Verlag, Berlin, 1997. \bibitem{i1} R. Ikehata and K. Tanizawa; \emph{Global existence of solutions for semilinear damped wave equations in $\mathbb{R}^N$ with noncompactly supported initial data}. Nonlinear Analysis \textbf{61} (2005), 1189-1208. \bibitem{j1} F. John; \emph{Blow-up of solutions of nonlinear wave equations in three space dimensions}. Manuscripta Math. \textbf{28}(1979), no. 1-3, 235-268. \bibitem{j2} F. John; \emph{Blow-up for quasilinear wave equations in three space dimensions}. Comm. Pure Appl. Math. \textbf{34}(1981), no. 1, 29-51. \bibitem{j3} F. John And S. Klainerman; \emph{Almost global existence to nonlinear wave equations in three space dimensions}. Comm. Pure Appl. Math. \textbf{37} (1984), no. 4, 443-455. \bibitem{j4} K. J\"{o}rgens; \emph{Das Anfangswertproblem im Grossen f\"{u}r eine Klasse nichtlinearer Wellengleichungen}. Math.Z. \textbf{77}(1961), 295-308 (German). \bibitem{k1} T. Kato; \emph{Blow-up of solutions of some nonlinear hyperbolic equations}. Comm. Pure Appl. Math. \textbf{33} (1980), no. 4, 501-505. \bibitem{k2} S. Kharibegashvili; \emph{On the existence or the absence of global solutions of the Cauchy characteristic problem for some nonlinear hyperbolic equations}. J. Boundary Value Problems \textbf{2005}(2005), no.3, 359-376. \bibitem{k3} S. Kharibegashvili; \emph{On the nonexistence of global solutions of the characteristic Cauchy problem for a nonlinear wave equation in a conical domain}. Differential Equations, \textbf{42}(2006), No.2, 279-290. Translated from Differential'nye Uravneniya, \textbf{42}(2006), No.2, 261-271. \bibitem{k4} S. Kharibegashvili; \emph{Some multidimensional problems for hyperbolic partial differential equations and systems}. Mem. Differential Equations Math. Phys. \textbf{37}(2006), 1-136. \bibitem{k5} M. A. Krasnosel'ski\v{\i}, P. P. Zabre\v{\i}ko, E. I. Pustyl'nik, and P. E. Sobolevski\v{\i}; \emph{Integral Operators in Spaces of Summable Functions}. Izdat. ``Nauka'', Moscow, 1966. \bibitem{k6} A. Kufner and S. Fuchik, Nonlinear Differential Equations. Izdat. "Nauka", Moscow, 1988. \bibitem{l1} O. A. Ladyzhenskaya; \emph{Boundary Value Problems of Mathematical Physics}. Izdat. ``Nauka", Moscow, 1973. \bibitem{l2} H. A. Levine; \emph{Instability and nonexistence of global solutions to nonlinear wave equations of the form $\textit{P}u_{tt}=-\textit{A}u+\mathfrak{F}(u)$}. Trans. Amer. Math. Soc. \textbf{192} (1974), 1-21. \bibitem{l3} L. Lundberg; \emph{The Klein-Gordon equation with light-cone data}. Comm. Math. Phys. \textbf{62}(1978), no.2, 107-118. \bibitem{m1} E. Mitidieri and S. I. Pohozaev; \emph{A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities}. (Russian) Trudy Mat. Inst. Steklova, \textbf{234}(2001), 1-384; English transl.: Proc. Steklov Inst. Math. 2001, no. 3(\textbf{234}), 1-362. \bibitem{s2} T. G. Sideris; \emph{Nonexistence of global solutions to semilinear wave equations in high dimensions}. J Differential Equations \textbf{52} (1984), no.3, 378-406. \bibitem{s3} W. A. Strauss; \emph{Nonlinear scattering theory at low energy}. J Funct. Anal. \textbf{41} (1981), no. 1, 110-133. \bibitem{t1} G. Todorova and E. Vitillaro; \emph{Blow-up for nonlinear dissipative wave equations in $\mathbb{R}^n$}. J. Math. Anal. Appl. \textbf{303} (2005), no.1, 242-257. \bibitem{t2} V. A. Trenogin; \emph{Functional Analysis}. Izdat. ``Nauka'', Moscow, 1993. \bibitem{y1} P.-F. Yao; \emph{Global smooth solutions for the quasilinear wave equation with boundary dissipation}. J. Differential Equations \textbf{241} (2007), 62-93. \bibitem{y2} B. Yordanov and Qi. S. Zhang; \emph{Finite time blow up for critical wave equations in high dimensions}. J. Funct. Anal. \textbf{231} (2006), 361-374. \bibitem{z1} J. Zhu; \emph{Blow-up of solutions of a semilinear hyperbolic equation and a parabolic equation with general forcing term and boundary condition}. Nonlinear Analysis \textbf{67} (2007), 33-38. \end{thebibliography} \end{document}