Electron. J. Diff. Equ., Vol. 2015 (2015), No. 211, pp. 1-10.

Critical exponent for a damped wave system with fractional integral

Mijing Wu, Shengjia Li, Liqing Lu

Abstract:
We shall present the critical exponent
$$ F(p, q,\alpha):=\max\big\{\alpha+\frac{(\alpha+1)(p+1)}{pq-1}, \alpha+\frac{(\alpha+1)(q+1)}{pq-1}\big\}-\frac{1}{2} $$
for the Cauchy problem
$$\displaylines{ u_{tt}-u_{xx}+u_t=J_{0|t}^{\alpha}(|v|^{p}), \quad (t,x)\in\mathbb{R}^{+}\times\mathbb{R},\cr v_{tt}-v_{xx}+v_t=J_{0|t}^{\alpha}(|u|^{q}), \quad (t, x)\in\mathbb{R}^{+}\times\mathbb{R},\cr (u(0,x), u_t(0,x))=(u_0(x),u_1(x)), \quad x\in \mathbb{R},\cr (v(0,x), v_t(0,x))=(v_0(x),v_1(x)), \quad x\in \mathbb{R},\cr }$$
where $p,q\geq 1$, $pq>1$ and $0<\alpha<1/2$; that is, the small data global existence of solutions can be derived to the problem above if $F(p, q, \alpha)<0$. Furthermore, in the case of $F(p, q, \alpha)\geq 0$ the non-existence of global solution can be obtained with the initial data having positive average value.

Submitted July 24, 2015. Published August 12, 2015.
Math Subject Classifications: 35B33.
Key Words: Damped wave equation; fractional integral; critical exponent; global solution.

Show me the PDF file (254 KB), TEX file, and other files for this article.

Mijing Wu
School of Mathematical Sciences
Shanxi University
Taiyuan, Shanxi 030006, China
email: mjwu@sxu.edu.cn
Shengjia Li
School of Mathematical Sciences
Shanxi University
Taiyuan, Shanxi 030006, China
email: sjli@sxu.edu.cn
Liqing Lu
School of Mathematical Sciences
Shanxi University
Taiyuan, Shanxi 030006, China
email: lulq@sxu.edu.cn

Return to the EJDE web page