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A priori estimates of solutions to local and nonlocal superlinear parabolic problems

Pavol Quittner

TL;DR

The paper develops a unified framework for a priori estimates of parabolic problems with local and nonlocal superlinear nonlinearities using energy methods, interpolation, bootstrap arguments, and Pohozaev-type identities. It proves AE and AE2 bounds in subcritical regimes, derives blow-up criteria via energy dynamics, and establishes continuity of the maximal existence time, while also showing the existence of nontrivial steady states for Choquard and related nonlocal models. The results encompass local, Choquard, Schrödinger–Poisson, and fractional Laplacian problems, with subcritical thresholds $p<p_S$, $p<p^*$, and $p<p_S(\alpha)$ guiding the applicability of the estimates. These contributions provide a robust toolkit for understanding blow-up behavior, energy dissipation, and steady-state multiplicity in a broad class of parabolic equations.

Abstract

We consider a priori estimates of possibly sign-changing solutions to superlinear parabolic problems and their applications (blow-up rates, energy blow-up, continuity of blow-up time, existence of nontrivial steady states etc). Our estimates are based mainly on energy, interpolation and bootstrap arguments, but we also use the Pohozaev identity, for example. We first discuss some known results on local problems and then consider problems with nonlocal nonlinearities or nonlocal differential operators. In particular, we deal with the fractional Laplacian and nonlinearities of Choquard type.

A priori estimates of solutions to local and nonlocal superlinear parabolic problems

TL;DR

The paper develops a unified framework for a priori estimates of parabolic problems with local and nonlocal superlinear nonlinearities using energy methods, interpolation, bootstrap arguments, and Pohozaev-type identities. It proves AE and AE2 bounds in subcritical regimes, derives blow-up criteria via energy dynamics, and establishes continuity of the maximal existence time, while also showing the existence of nontrivial steady states for Choquard and related nonlocal models. The results encompass local, Choquard, Schrödinger–Poisson, and fractional Laplacian problems, with subcritical thresholds , , and guiding the applicability of the estimates. These contributions provide a robust toolkit for understanding blow-up behavior, energy dissipation, and steady-state multiplicity in a broad class of parabolic equations.

Abstract

We consider a priori estimates of possibly sign-changing solutions to superlinear parabolic problems and their applications (blow-up rates, energy blow-up, continuity of blow-up time, existence of nontrivial steady states etc). Our estimates are based mainly on energy, interpolation and bootstrap arguments, but we also use the Pohozaev identity, for example. We first discuss some known results on local problems and then consider problems with nonlocal nonlinearities or nonlocal differential operators. In particular, we deal with the fractional Laplacian and nonlinearities of Choquard type.

Paper Structure

This paper contains 8 sections, 8 theorems, 149 equations, 1 table.

Table of Contents

  1. Introduction and main results
  2. Early results and applications to blow-up rates and decay
  3. Generalizations and further applications: Local problems
  4. Nonlocal problems
  5. Proofs
  6. Auxiliary results
  7. Proofs of Theorem \ref{['thm23']} and Corollary \ref{['cor23']}
  8. Proofs of Theorems \ref{['thmChoquard']} and \ref{['thmFracLap']}

Key Result

Theorem 1.1

Let $q\in(2,5)$, $\lambda>0$, $\Omega\subset{\mathbb R}^3$ be bounded and smooth, and let $u_0\in X$, where $X=L_R(\Omega)$ and $R\in(\frac{3}{2}(q-1)\max(1,\frac{3}{q}),\infty]$. If $q\le3$, then assume also that $\Omega$ is starshaped. Let $u$ be the solution of pbmSPS with the maximal existence t Then (ii) Assume that E-pos fails. Then In addition, if $q\ge3$, then

Theorems & Definitions (20)

  • Remark 1.1
  • Theorem 1.1
  • Corollary 1.1
  • Remark 1.2
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • ...and 10 more