17,652 research outputs found
Nonlinear Convection in Reaction-diffusion Equations under dynamical boundary conditions
We investigate blow-up phenomena for positive solutions of nonlinear
reaction-diffusion equations including a nonlinear convection term in a bounded domain of
under the dissipative dynamical boundary conditions . Some conditions on and are discussed
to state if the positive solutions blow up in finite time or not. Moreover, for
certain classes of nonlinearities, an upper-bound for the blow-up time can be
derived and the blow-up rate can be determinated.Comment: 20 page
Non-self-similar blow-up in the heat flow for harmonic maps in higher dimensions
We analyze the finite-time blow-up of solutions of the heat flow for
-corotational maps . For each dimension
we construct a countable family of blow-up solutions via a
method of matched asymptotics by glueing a re-scaled harmonic map to the
singular self-similar solution: the equatorial map. We find that the blow-up
rates of the constructed solutions are closely related to the eigenvalues of
the self-similar solution. In the case of -corotational maps our solutions
are stable and represent the generic blow-up.Comment: 26 pages, 5 figure
A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions
In this paper we study a simple non-local semilinear parabolic equation with
Neumann boundary condition. We give local existence result and prove global
existence for small initial data. A natural non increasing in time energy is
associated to this equation. We prove that the solution blows up at finite time
if and only if its energy is negative at some time before . The proof of
this result is based on a Gamma-convergence technique
On nonexistence of Baras--Goldstein type for higher-order parabolic equations with singular potentials
An analogy of nonexistence result by Baras and Goldstein (1984), for the heat
equation with inverse singular potential, is proved for 2mth-order linear
parabolic equations with Hardy-supercritical singular potentials. Extensions to
other linear and nonlinear singular PDEs are discussed.Comment: 22 page
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