19 research outputs found
On the uniqueness of bounded solutions to singular parabolic problems
No full textWe provide criteria for uniqueness or nonuniqueness of bounded solutions for a wide class of second order parabolic problems with singular coefficients
Sublinear elliptic problems with a Hardy potential
No full textIn this paper we study the positive solutions of sub linear elliptic equations with a Hardy potential which is singular at the boundary. By means of ODE techniques a fairly complete picture of the class of radial solutions is given.
Local solutions with a prescribed growth at the boundary are constructed by means of contraction operators. Some of those radial solutions are then used to construct ordered upper and lower solutions in general domains. By standard iteration arguments the existence of positive solutions is proved. An important tool is the Hardy constant
Dynamical Structure of Some Nonlinear Degenerate Diffusion Equations
No full textWe consider degenerate reaction diffusion equations of the form u(t) = Delta u(m) + f(x, u), where f (x, u) similar to a(x) u(p) with 1 0 at least in some part of the spatial domain, so that u = 0 is an unstable stationary solution. We prove that the unstable manifold of the solution u = 0 has infinite Hausdorff dimension, even if the spatial domain is bounded. This is in marked contrast with the case of non-degenerate semilinear equations. The above result follows by first showing the existence of a solution that tends to 0 as t -> -infinity while its support shrinks to an arbitrarily chosen point x* in the region where a(x) > 0, then superimposing such solutions, to form a family of solutions of arbitrarily large number of free parameters. The construction of such solutions will be done by modifying self-similar solutions for the case where a is a constant
Parabolic equations with non linear, degenerate and space-time dependent operators
No full textIn this paper existence and regularity results for a class of degenerate nonlinear parabolic equations are proved. Indeed, the diffusion operator may degenerate as the solution diverges and may depend on space and time variables in a non–regular way, too. Some estimates on the behaviour of the solutions for diverging t are also given
The Fujita exponent for the Cauchy problem in the hyperbolic space
No full textIt is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the Euclidean space. The main purpose of this paper is to study its effect on the positive solutions of Cauchy problems with power nonlinearities. Existence and non-existence results for local solutions are derived. Emphasis is put on their long time behavior and on Fujita's phenomenon. To have the same situation as for the Cauchy problem in R-N, namely finite time blow up for all solutions if the exponent is smaller than a critical value and existence of global solutions only for powers above the critical exponent, we must introduce a weight depending exponentially on the time. In this respect the situation is similar to problems in bounded domains with Dirichlet boundary conditions. Important tools are estimates for the heat kernel in the hyperbolic space and comparison principles. (C) 2011 Published by Elsevier Inc
