Electron. J. Differential Equations, Vol. 2017 (2017), No. 227, pp. 1-42.

Differential equations with a difference quotient

Brian Street

Abstract:
The purpose of this paper is to study a class of ill-posed differential equations. In some settings, these differential equations exhibit uniqueness but not existence, while in others they exhibit existence but not uniqueness. An example of such a differential equation is, for a polynomial $P$ and continuous functions $f(t,x):[0,1]\times [0,1]\to \mathbb{R}$,
$$ \frac{\partial}{\partial t} f(t,x) = \frac{ P(f(t,x))-P(f(t,0))}{x}, \quad x>0. $$
These differential equations are related to inverse problems.

Submitted January 7, 2017. Published September 21, 2017.
Math Subject Classifications: 34K29, 34K09.
Key Words: Ill-posed; differential equation with difference quotient; existence without uniqueness; uniqueness without existence; inverse problem.

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Brian Street
University of Wisconsin-Madison
Department of Mathematics
480 Lincoln Dr.
Madison, WI 53706, USA
email: street@math.wisc.edu

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