Nonexistence of Global Weak Solutions of Nonlinear Keldysh Type Equation with One Derivative Term

Kangqun Zhang

doi:10.1155/2018/3931297

Abstract

We focus on the nonexistence of global weak solutions of nonlinear Keldysh type equation with one derivative term. In terms of the analysis of the first Fourier coefficient, we show the solution of singular initial value problem and singular initial-boundary value problem of the nonlinear equation with positive initial data blow-up in some finite time interval.

Highlights

In this paper we consider the nonexistence of global weak solution of nonlinear Keldysh type equation (1) with modified initial data (2) and boundary value (3); that is, t∂t2u − △u + b∂tu = f (u), in (0, T) × Ω (1)u (0, x) = φ1 (x), lim t󳨀→0 tb∂tu (t, x) = φ2 (x) (2)in Ω and u (t, x) = 0 on (0, T) × ∂Ω (3)where △ = ∑ni=1 ∂x2i is the Laplace differential operator and b ∈ (0, 1) is a parameter
We focus on the nonexistence of global weak solutions of nonlinear Keldysh type equation with one derivative term
In terms of the analysis of the first Fourier coefficient, we show the solution of singular initial value problem and singular initial-boundary value problem of the nonlinear equation with positive initial data blow-up in some finite time interval

Summary

Introduction

In this paper we consider the nonexistence of global weak solution of nonlinear Keldysh type equation (1) with modified initial data (2) and boundary value (3); that is, t∂t2u − △u + b∂tu = f (u) , in (0, T) × Ω (1). For nonlinear wave equation with constant coefficients, there exist many well-known results including existence and blow-up of solutions. In order to show the first Fourier coefficient of the solution is positive and approaches infinity at finite time, we derive a nonlinear second-order singular differential inequality:. Based on the cautious calculation of singular differential inequality (5), we can deal with the nonexistence of global solution to the nonlinear problem and the proof is independent of the weak conservation of energy law and concavity of argument.

Singular Initial-Boundary Value Problem

Singular Initial Value Problem