Numerical solution of a fast diffusion equation

Abstract

Abstract. In this paper, the authors consider the first boundary value problem for the nonlinear reaction diffusion equation: ut − ∆u m = αu p 1 in Ω, a smooth bounded domain in R d (d ≥ 1) with the zero lateral boundary condition and with a positive initial condition, m ∈ ]0, 1[ (fast diffusion problem), α ≥ 0 and p 1 ≥ m. Sufficient conditions on the initial data are obtained for the solution to vanish or become infinite in a finite time. A scheme for the discretization in time of this problem is proposed. The numerical scheme preserves the essential properties of the initial problem; namely existence of an extinction or a blow-up time, for which estimates have been obtained. The convergence of the method is also proved.

Nontrivial, nonnegative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms

“…√ 5 2 , 2 only the slow diffusion is possible under the condition m > p. In the case of the fast diffusion and superlinear growth in u, v of the right hand sides, the solutions may blow up or vanish in some finite time depending on the initial conditions as illustrated in [11], [40], [41], [46] and the references therein for the simple equation obtained from the first of system (1.1) by letting p = 2, a > 0 constant and all the kernels K i ≡ 0 (observe that in the case when p = q = 2 no restriction on m, n are required, see Remark 2.2). If Ω = R N for such equation we have that the solution blows up for any initial condition in the case when the superlinear growth in u is less than a certain critical exponent, see [46], and the same occurs for doubly degenerate parabolic equations, see [47].…”

“…If Ω = R N for such equation we have that the solution blows up for any initial condition in the case when the superlinear growth in u is less than a certain critical exponent, see [46], and the same occurs for doubly degenerate parabolic equations, see [47]. If the growth is linear or sublinear we do not have blow up of any solution, see [41], hence solutions exist for all t ≥ 0 and in the linear case, depending on the initial condition, they may vanish in finite time or become unbounded as t → +∞ and, thus, the considered initial conditions cannot give rise to a periodic solution.…”

Discretization of the Cauchy problem for a fast diffusion equation

“…Recently, a numerical method was proposed in [14] to solve this problem in the special case p 1 = m + 1. An implicit time-discretization was also investigated in [15][16][17] for solving analogous fast and slow diffusion problems on a bounded domain with zero lateral boundary condition and positive initial data. The proposed scheme preserves the essential properties of the initial problem, namely the (possibly) existence of an extinction or a finite blow up time.…”

Finite element approximation for degenerate parabolic equations. an application of nonlinear semigroup theory

“…Those works were done in the literature of porous media or that of Stefan problems. For fast diffusion problems, we refer to [23,24].…”