Blow-Up of Positive Solutions to Wave Equations in High Space Dimensions

This paper is concerned with the Cauchy problem for the semilinear wave equation: u_{tt}-\Delta u=F(u) \ \mbox{in} \ R^n\times[0, \infty), where the space dimension $n \ge 2$, $F(u)=|u|^p$ or $F(u)=|u|^{p-1}u$ with $p>1$. Here, the Cauchy data are non-zero and non-compactly supported. Our results on the blow-up of positive radial solutions (not necessarily radial in low dimensions $n=2, 3$) generalize and extend the results of Takamura(1995) and Takamura, Uesaka and Wakasa(2011). The main technical difficulty in the paper lies in obtaining the lower bounds for the free solution when both initial position and initial velocity are non-identically zero in even space dimensions.Comment: 16page

Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping

Presented here is a study of a viscoelastic wave equation with supercritical source and damping terms. We employ the theory of monotone operators and nonlinear semigroups, combined with energy methods to establish the existence of a unique local weak solution. In addition, it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense.Comment: The 2nd version includes a new proof of the energy identit

Global Existence and Non-existence Theorems for Nonlinear Wave Equations

In this article we focus on the global well-posedness of an initial-boundary value problem for a nonlinear wave equation in all space dimensions. The nonlinearity in the equation features the damping term |u|k |ut|m sgn(ut) and a source term of the form |u|p-1u, where k, p ≥ 1 and 0 \u3c m \u3c 1. In addition, if the space dimension n ≥ 3, then the parameters k, m and p satisfy p, k/(1-m) ≤ n/(n - 2). We show that whenever k + m ≥ p, then local weak solutions are global. On the other hand, we prove that whenever p \u3e k + m and the initial energy is negative, then local weak solutions blow-up in finite time, regardless of the size of the initial data

On the paper "Symmetry analysis of wave equation on sphere" by H. Azad and M. T. Mustafa

Using the scalar curvature of the product manifold S^{2}X R and the complete group classification of nonlinear Poisson equation on (pseudo) Riemannian manifolds, we extend the previous results on symmetry analysis of homogeneous wave equation obtained by H. Azad and M. T. Mustafa [H. Azad and M. T. Mustafa, Symmetry analysis of wave equation on sphere, J. Math. Anal. Appl., 333 (2007) 1180--1888] to nonlinear Klein-Gordon equations on the two-dimensional sphere.Comment: Version accepted in J. Math. Anal. App

The Glassey conjecture with radially symmetric data

In this paper, we verify the Glassey conjecture in the radial case for all spatial dimensions, which states that, for the nonlinear wave equations of the form $\Box u=|\nabla u|^p$, the critical exponent to admit global small solutions is given by $p_c=1+\frac{2}{n-1}$. Moreover, we are able to prove the existence results with low regularity assumption on the initial data and extend the solutions to the sharp lifespan. The main idea is to exploit the trace estimates and KSS type estimates.Comment: 28 page