177 research outputs found
Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified Groups
In this paper, we prove a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified Lie groups. Our proof is based on the concavity argument and the Poincar\'e inequality, established in [38] for stratified groups
Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source
This paper deals with the long-time behavior of solutions of nonlinear
reaction-diffusion equations describing formation of morphogen gradients, the
concentration fields of molecules acting as spatial regulators of cell
differentiation in developing tissues. For the considered class of models, we
establish existence of a new type of ultra-singular self-similar solutions.
These solutions arise as limits of the solutions of the initial value problem
with zero initial data and infinitely strong source at the boundary. We prove
existence and uniqueness of such solutions in the suitable weighted energy
spaces. Moreover, we prove that the obtained self-similar solutions are the
long-time limits of the solutions of the initial value problem with zero
initial data and a time-independent boundary source
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