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On $L^{\infty }$-estimates and the structure of the global attractor for weak solutions of reaction-diffusion equations

Rubén Caballero, Piotr Kalita, José Valero

Abstract

In this paper, we study the structure of the global attractor for weak and regular solutions of a problem governed by a scalar semilinear reaction-diffusion equation with a non-regular nonlinearity, such that uniquness of solutions can fail to happen. First, using the Moser--Alikakos iterations we obtain the estimates of the weak solutions in the space $L^{\infty }(Ω)$. After that, using these estimates we improve the existing results on the structure of the attractor. Finally, estimates of the Hausdorff and fractal dimension of the attractor are obtained.

On $L^{\infty }$-estimates and the structure of the global attractor for weak solutions of reaction-diffusion equations

Abstract

In this paper, we study the structure of the global attractor for weak and regular solutions of a problem governed by a scalar semilinear reaction-diffusion equation with a non-regular nonlinearity, such that uniquness of solutions can fail to happen. First, using the Moser--Alikakos iterations we obtain the estimates of the weak solutions in the space . After that, using these estimates we improve the existing results on the structure of the attractor. Finally, estimates of the Hausdorff and fractal dimension of the attractor are obtained.
Paper Structure (10 sections, 14 theorems, 138 equations)

This paper contains 10 sections, 14 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Semilinear heat equation: problem setup and previous results in KKV14KKV15KV06KV09.
  3. Attractor for weak solutions: existence
  4. Regular solutions: existence and structure of the attractor
  5. Weak solutions: structure of the attractor
  6. Moser--Alikakos iterations and $L^\infty$ estimates for weak solutions
  7. Existence and structure of the global attractor for weak and regular solutions
  8. Uniqueness and estimate of the fractal dimension
  9. Uniqueness and continuity on attractor.
  10. Upper bound on the attractor dimension.

Key Result

Theorem 3.1

If $u$ is a weak solution of (Eq) with $\sup_{s\geq t_{1}}\Vert u(s)\Vert _{2}<\infty$, then for every $\tau >0$ there exists a constant $D(\tau )$ such that for every $t\geq t_{1}+\tau$ we have $u(t)\in L^{\infty }(\Omega)$ and

Theorems & Definitions (23)

  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • ...and 13 more