Parabolic Systems in Unbounded Domains I. Existence and Dynamics

Abstract

In this paper we investigate the existence, uniqueness, and asymptotic behavior of a solution for a class of coupled nonlinear parabolic equations in a general unbounded domain that includes the whole space ‫ޒ‬ n , the exterior of a bounded domain, and a half space in ‫ޒ‬ n . The asymptotic behavior of the solution is with respect to a pair of quasi-solutions of the corresponding elliptic system, and when these two quasi-solutions coincide the solution of the parabolic system converges to a unique solution of the elliptic system. ᮊ

An application of the monotone iterative method to a combustion problem in porous media

“…By Lemmas 1-3, the sequences U (k) , U (k) are well defined and we can apply [6,Theorem 3.1] to conclude the first statement of Theorem 2. The second statement, i.e.…”

“…In this section, for the sake of completeness and clarity we briefly describe the method, stating the main results we need in the next section. For more details we refer the reader to [4,5] (see also [6]). …”

Identification of the Source for Full Parabolic Equations

“…Note that this is a linear parabolic equations with constant coefficients. The existence and uniqueness of the solution to (2.1) is discussed in [29]. It is assumed that u(•, t), f (•) ∈ L 2 (R n ) are unknown functions and that y ∈ L 2 (R n ) can be measured with certain noise level δ, i.e., the data function…”

Spreading speeds along shifting resource gradients in reaction-diffusion models and lattice differential equations.

“…Since 1 is an arbitrary small number, (78) and (77) show that Q[u 2 ](t, x) ≤ u 2 (t, x). It follows that…”