Existence and regularity results for terminal value problem for nonlinear fractional wave equations

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions for terminal value problems in nonlinear time-fractional wave equations with Caputo derivatives, including applications to specific models like Ginzburg-Landau and Burgers equations.

Contribution

It establishes well-posedness and regularity results for a broad class of nonlinear fractional wave equations with Caputo derivatives, extending previous work to include inverse problems.

Findings

Proved existence and uniqueness of solutions.

Derived regularity results for solutions and derivatives.

Applied methods to Ginzburg-Landau and Burgers models.

Abstract

We consider the terminal value problem (or called final value problem, initial inverse problem, backward in time problem) of determining the initial value, in a general class of time-fractional wave equations with Caputo derivative, from a given final value. We are concerned with the existence, regularity of solutions upon the terminal value. Under several assumptions on the nonlinearity, we address and show the well-posedness (namely, the existence, uniqueness, and continuous dependence) for the terminal value problem. Some regularity results for the mild solution and its derivatives of first and fractional orders are also derived. The effectiveness of our methods are showed by applying the results to two interesting models: Time fractional Ginzburg-Landau equation, and Time fractional Burgers equation, where time and spatial regularity estimates are obtained.