Lifespan estimate for one dimensional wave equation with semilinear terms of spatial derivative

TL;DR

This paper investigates the lifespan bounds of solutions to one-dimensional wave equations with semilinear terms involving spatial derivatives, highlighting new proof techniques and connections to classical differential inequalities.

Contribution

It provides new upper and lower lifespan bounds for solutions with space-derivative nonlinearities and links these results to known inequalities for damped wave equations.

Findings

Sharp upper bound of lifespan obtained.

Part of the bounds matches classical differential inequalities.

Simple proof for blow-up results using iteration and slicing methods.

Abstract

This paper studies the upper and lower bounds of the lifespan for the classical solutions to the initial value problems of one dimensional wave equations with non-autonomous semilinear terms including the space-derivative of the unknown function.This is a non-trivial business comparing to the analogous results with time-derivative type semilinear terms, especially for the proof to obtain the sharp upper bound of the lifespan as we have to deal with space dependent weights among iteration procedures of the weighted functional of the solution. Also it is surprising that a part of them reaches to the same ordinary differential inequality for classical semilinear damped wave equations introduced by Li and Zhou (Discrete Contin. Dynam. Systems, 1995, 1(4): 503-520), and we show a simple proof for blow up result from this ordinary differential inequality by iteration argument and slicing method in more general situation.