2,411 research outputs found
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
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
Global Regularity vs. Finite-Time Singularities: Some Paradigms on the Effect of Boundary Conditions and Certain Perturbations
In light of the question of finite-time blow-up vs. global well-posedness of
solutions to problems involving nonlinear partial differential equations, we
provide several cautionary examples which indicate that modifications to the
boundary conditions or to the nonlinearity of the equations can effect whether
the equations develop finite-time singularities. In particular, we aim to
underscore the idea that in analytical and computational investigations of the
blow-up of three-dimensional Euler and Navier-Stokes equations, the boundary
conditions may need to be taken into greater account. We also examine a
perturbation of the nonlinearity by dropping the advection term in the
evolution of the derivative of the solutions to the viscous Burgers equation,
which leads to the development of singularities not present in the original
equation, and indicates that there is a regularizing mechanism in part of the
nonlinearity. This simple analytical example corroborates recent computational
observations in the singularity formation of fluid equations
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