Quenching time for a system of semilinear heat equations

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...

Feedback Stabilization of Semilinear Heat Equations

Holomorphic Solutions of Semilinear Heat Equations

with φ ∈ L(R), where P is a polynomial vanishing at the origin and ∆ stands for the Laplacian with respect to x. The analyticity in time of the solutions of a semilinear heat equation has been considered by many authors. For example Ōuchi [2] treated the analyticity in time of the solutions of certain initial boundary value problems with bounded continuous initial functions, which include (1) i...

Singular perturbations to semilinear stochastic heat equations

We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter ε tends to zero, their solutions converge to the ‘wrong’ limit, i.e. they do not converge to the solution obtained by simply setting ε = 0. A similar effect is also observed for some (formally) small stocha...

Quenching for semidiscretizations of a semilinear heat equation with Dirichlet and Neumann boundary conditions

This paper concerns the study of the numerical approximation for the following boundary value problem: 8><>: ut(x, t) − uxx(x, t) = −u(x, t), 0 < x < 1, t > 0, ux(0, t) = 0, u(1, t) = 1, t > 0, u(x, 0) = u0(x) > 0, 0 ≤ x ≤ 1, where p > 0. We obtain some conditions under which the solution of a semidiscrete form of the above problem quenches in a finite time and estimate its semidiscrete quenchi...