Existence and nonexistence of global solutions to fast diffusions with source

Abstract

ABSTRACT. We consider positive solutions to the Cauchy problem for quasilinear parabolic equations dtu = Au 171 + u p with max{0,1 -2/N} < m < 1 < p, where N is the space dimension. Putting p^ =m + 2/N, we shall show that if p < p^, then all nontrivial solutions blow up in finite time, and if p > p^, then there are nontrivial global solutions.

The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms

“…For example, the author of [12] shows that p c = 1 + σ + 2/n is the critical exponent of the problem u t − u 1+σ = u p in R n . Later, the authors of [11] show that p c belongs to the blow-up case. The authors of [1,9] discuss the more general equations and the doubly singular parabolic equations, respectively, and obtain similar results.…”

Life span and a new critical exponent for a degenerate parabolic equation

“…In [4,6,19] they have proved that the solution u(x, t) of (1.2) blow up if 1 < p < p * m ; while both global and nonglobal positive solutions exist if p > p * m . When p = m + 2 N , in [16,17] Mochizuki, Mukai and Suzuki have proved that the solutions of (1.2) blow up in finite time (see also [5,7,20]). For more references on this topic, we refer the readers to see [1,13,14] and references therein.…”

Discretization of the Cauchy problem for a fast diffusion equation

“…Most of the analysis of (1.1) is known, with the proper changes for equations with no sources, that is = 0 (see [1,10,13]). Numerous papers were also devoted to the case when > 0 (see [9,19,21,22]). A main source of interests for (1.1) lies in the remarkable properties of its solutions, such as blow-up in finite time as well as infinite propagation of disturbances.…”