Blow-up of a nonlocal semilinear parabolic equation with positive initial energy

Blow-up of a nonlocal semilinear parabolic equation with positive initial energy

Blow up of Solutions with Positive Initial Energy for the Nonlocal Semilinear Heat Equation

In this paper, we investigate a nonlocal semilinear heat equation with homogeneous Dirichlet boundary condition in a bounded domain, and prove that there exist solutions with positive initial energy that blow up in finite time.

Blow-up for a Semilinear Parabolic Equation with Nonlinear Memory and Nonlocal Nonlinear Boundary

where Ω is a bounded domain in RN for N ≥ 1 with C2 boundary ∂Ω, p, q, l and k are positive parameters, the weight function f (x, y) is nonnegative, nontrivial, continuous and defined for x ∈ ∂Ω, y ∈ Ω, while the nonnegative nontrivial initial Received November 14, 2012, accepted January 28, 2013. Communicated by Eiji Yanagida. 2010 Mathematics Subject Classification: 35B35, 35K50, 35K55.

Blow-up analysis for a semilinear parabolic equation with nonlinear memory and nonlocal nonlinear boundary condition

In this paper, we consider a semilinear parabolic equation ut = ∆u + u q ∫ t 0 u(x, s)ds, x ∈ Ω, t > 0 with nonlocal nonlinear boundary condition u|∂Ω×(0,+∞) = ∫ Ω φ(x, y)u (y, t)dy and nonnegative initial data, where p, q ≥ 0 and l > 0. The blow-up criteria and the blow-up rate are obtained.

Interior Gradient Blow-up in a Semilinear Parabolic Equation

We present a one dimensional semilinear parabolic equation for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. In our example the derivative blows up in the interior of the space interval rather than at the boundary, as in earlier examples. In the case of monotone solutions we show that gradient blow-up occurs at a single...