Backward selfsimilar solutions of supercritical parabolic equations

Backward selfsimilar solutions of supercritical parabolic equations

We consider the exponential reaction–diffusion equation in space-dimension n ∈ (2, 10). We show that for any integer k ≥ 2 there is a backward selfsimilar solution which crosses the singular steady state k-times. The sameholds for the power nonlinearity if the exponent is supercritical in the Sobolev sense and subcritical in the Joseph–Lundgren sense. © 2008 Elsevier Ltd. All rights reserved.

Backward Uniqueness for Parabolic Equations

It is shown that a function u satisfying |∂t + u| M (|u| + |∇u|), |u(x, t)| MeM|x| in (R \ BR) × [0, T ] and u(x, 0) = 0 for x ∈ R \ BR must vanish identically in R \ BR × [0, T ].

Homogenization of forward-backward parabolic equations

We study the homogenization of the equation R(εx) ∂uε ∂t −∆uε = f , where R is a periodic function which may vanish or change sign, with appropriate initial/final conditions. The main tool is a compactness result for sequences of functions which have bounded norms in the spaces associated to the problems.

Linear Stability of Selfsimilar Solutions of Unstable Thin-film Equations

We study the linear stability of selfsimilar solutions of long-wave unstable thin-film equations with power-law nonlinearities ut = −(u nuxxx + u mux)x 0 < n < 3, n ≤ m Steady states, which exist for all values of m and n above, are shown to be stable if m ≤ n + 2 when 0 < n ≤ 2, marginally stable if m ≤ n + 2 when 2 < n < 3 and unstable otherwise. Dynamical selfsimilar solutions are known to e...

Recent Results on Selfsimilar Solutions of Degenerate Nonlinear Diiusion Equations