Reaction–Diffusion in a Closed Domain Formed by Irregular Curves

Journal of Mathematical Analysis and Applications

2001

We study the Dirichlet problem for the parabolic equation u t = u m m > 0, in a bounded, non-cylindrical and non-smooth domain ⊂ N+1 N ≥ 2. Existence and boundary regularity results are established. We introduce a notion of parabolic modulus of left-lower (or left-upper) semicontinuity at the points of the lateral boundary manifold and show that the upper (or lower) Hölder condition on it plays a crucial role for the boundary continuity of the constructed solution. The Hölder exponent 1 2 is critical as in the classical theory of the one-dimensional heat equation u t = u xx .

Section: Introductionmentioning

confidence: 96%

On the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains

Journal of Mathematical Analysis and Applications

2001

“…In many cases this may be non-smooth and characteristic. It should be mentioned that in the one-dimensional case Dirichlet and Cauchy-Dirichlet problems for the reaction-diffusion equations in irregular domains were studied in recent papers by the author [3,4]. Primarily applying this theory a complete description of the evolution of interfaces were presented in other recent papers [5,6].…”

Section: Introductionmentioning

confidence: 99%

Well-posedness of the Dirichlet problem for the non-linear diffusion equation in non-smooth domains

Trans. Amer. Math. Soc.

2004

Self Cite

Abstract. We investigate the Dirichlet problem for the parablic equation Math. Anal. Appl., 260, 2 (2001), 384-403] existence and boundary regularity results were established. In this paper we present uniqueness and comparison theorems and results on the continuous dependence of the solution on the initial-boundary data. In particular, we prove L 1 -contraction estimation in general non-smooth domains.

“…In many cases this may be a characteristic single point. It should be mentioned that in the one-dimensional case Dirichlet and Cauchy-Dirichlet problems for the reaction-diffusion equations in irregular domains were studied in papers by the author [11,12]. Primarily applying this theory a complete description of the evolution of interfaces were presented in other papers [13,14].…”

Section: Boundary Value Problemsmentioning

confidence: 99%

“…Since the uniqueness and comparison results of this paper significantly improve the one-dimensional results from [11,12], we describe the one-dimensional results separately in Section 3. We prove Theorems 2.2, 2.6, and 2.7 in Sections 4-6, respectively.…”

Section: Boundary Value Problemsmentioning

confidence: 99%

Reaction-Diffusion in Nonsmooth and Closed Domains

Boundary Value Problems

2007

Self Cite

We investigate the Dirichlet problem for the parabolic equationin a nonsmooth and closed domain Ω ⊂ R N+1 , N ≥ 2, possibly formed with irregular surfaces and having a characteristic vertex point. Existence, boundary regularity, uniqueness, and comparison results are established. The main objective of the paper is to express the criteria for the well-posedness in terms of the local modulus of lower semicontinuity of the boundary manifold. The two key problems in that context are the boundary regularity of the weak solution and the question whether any weak solution is at the same time a viscosity solution.