Quenching Time for Some Semilinear Equations with A Potential

Some Limiting Situations for Semilinear Elliptic Equations

— The objective of this mini-course is to take a look at a standard semilinear partial differential equation −∆u = λf(u) on which we show the use of some basic tools in the study of elliptic equation. We will mention the maximum principle, barrier method, blow-up analysis, regularity and boot-strap argument, stability, localization and quantification of singularities, Pohozaev identities, movin...

Singular Solutions for some Semilinear Elliptic Equations

We are concerned with the behavior of u near x = O. There are two distinct cases: 1) When p >= N / ( N -2) and (N ~ 3) it has been shown by BR~ZIS & V~RON [9] that u must be smooth at 0 (See also BARAS & PIERRE [1] for a different proof). In other words, isolated singularities are removable. 2) When 1-< p < N / ( N 2) there are solutions of (1) with a singularity at x ---0. Moreover all singula...

The Quenching of Solutions of Semilinear Hyperbolic Equations

We consider the problem u, U,,x + b (u(x, t)), 0< x < L, >0; u(0, t) u(L, t)=0; u(x, O) ut(x, 0)=0. Assume that b (-oe, A) (0, ee) is continuously differentiable, monotone increasing, convex, and satisfies lim,_.a-b(u)= +ee. We prove that there exist numbers L1 and L2, 0 L2, then a weak solution u (to be defined) quenches in the sense that u reaches A in finite time; if...

Critical Exponent for Semilinear Wave Equations with Space-Dependent Potential

We study the balance between the effect of spatial inhomogeneity of the potential in the dissipative term and the focusing nonlinearity. Sharp critical exponent results will be presented in the case of slow decaying potential.

Quenching Time for a Nonlocal Diffusion Problem with Large Initial Data