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Regularizing effect and decay results for a parabolic problem with repulsive superlinear first order terms

Martina Magliocca

TL;DR

This work studies a parabolic equation with a repulsive superlinear gradient term in divergence form, allowing unbounded initial data and rough coefficients. It develops a nonlinear renormalized-solution framework and a delta-argument to establish sharp regularizing effects from L^σ or L^1 data to higher L^r spaces, together with explicit time-decay rates. Key contributions include L^σ–L^σ contraction, L^σ–L^r smoothing with precise decay exponents, and long-time decay (including L^∞ bounds and extinction in certain regimes), plus analogous results for L^1 data via renormalized and Marcinkiewicz regularities. The results extend classical regularity/decay theory to degenerate parabolic problems with gradient-source terms, under Leray–Lions structure and minimal regularity on A(t,x) and initial data, with implications for understanding smoothing, decay, and robustness to rough data in nonlinear diffusion problems.

Abstract

We want to analyse both regularizing effect and long, short time decay concerning parabolic Cauchy-Dirichlet problems of the type \begin{equation*} \begin{cases} \begin{array}{ll} u_t-\text{div} (A(t,x)|\nabla u|^{p-2}\nabla u)=γ|\nabla u|^q & \text{in}\,\,Q_T,\\ u=0 &\text{on}\,\,(0,T)\times\partialΩ,\\ u(0,x)=u_0(x) &\text{in}\,\, Ω. \end{array} \end{cases} \end{equation*} We assume that $A(t,x)$ is a coercive, bounded and measurable matrix, the growth rate $q$ of the gradient term is superlinear but still subnatural, $γ>0$, the initial datum $u_0$ is an unbounded function belonging to a well precise Lebesgue space $L^σ(Ω)$ for $σ=σ(q,p,N)$.

Regularizing effect and decay results for a parabolic problem with repulsive superlinear first order terms

TL;DR

This work studies a parabolic equation with a repulsive superlinear gradient term in divergence form, allowing unbounded initial data and rough coefficients. It develops a nonlinear renormalized-solution framework and a delta-argument to establish sharp regularizing effects from L^σ or L^1 data to higher L^r spaces, together with explicit time-decay rates. Key contributions include L^σ–L^σ contraction, L^σ–L^r smoothing with precise decay exponents, and long-time decay (including L^∞ bounds and extinction in certain regimes), plus analogous results for L^1 data via renormalized and Marcinkiewicz regularities. The results extend classical regularity/decay theory to degenerate parabolic problems with gradient-source terms, under Leray–Lions structure and minimal regularity on A(t,x) and initial data, with implications for understanding smoothing, decay, and robustness to rough data in nonlinear diffusion problems.

Abstract

We want to analyse both regularizing effect and long, short time decay concerning parabolic Cauchy-Dirichlet problems of the type \begin{equation*} \begin{cases} \begin{array}{ll} u_t-\text{div} (A(t,x)|\nabla u|^{p-2}\nabla u)=γ|\nabla u|^q & \text{in}\,\,Q_T,\\ u=0 &\text{on}\,\,(0,T)\times\partialΩ,\\ u(0,x)=u_0(x) &\text{in}\,\, Ω. \end{array} \end{cases} \end{equation*} We assume that is a coercive, bounded and measurable matrix, the growth rate of the gradient term is superlinear but still subnatural, , the initial datum is an unbounded function belonging to a well precise Lebesgue space for .

Paper Structure

This paper contains 11 sections, 15 theorems, 146 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Assumptions
  3. The growth range with $L^\sigma(\Omega)$ data
  4. $L^{\sigma}(\Omega)-L^{\sigma}(\Omega)$ regularity
  5. The regularizing effect $L^{\sigma}(\Omega)-L^{r}(\Omega)$
  6. Long time decay results
  7. The growth range with $L^1(\Omega)$ data
  8. $L^1-L^1$ and Marcinkiewicz regularities
  9. The regularizing effect $L^1-L^r$ and long time decays
  10. Further comments
  11. On the notion of solution

Key Result

Lemma 3.2

Assume (ID1), (A1)--(A2) with $p>\frac{2N}{N+\sigma}$ and (H0) with (Q1). Moreover, let $u$ be a solution of (P) in the sense of defrin1. Then, there exists a positive value $\delta_0$ such that, for every $k>0$ and for every $\delta<\delta_0$ satisfying we have

Figures (5)

  • Figure 1: Colours legend
  • Figure 2: The case $2\le p<N$
  • Figure 3: The case $\frac{2N}{N+1}<p< 2$
  • Figure 4: The case $\frac{2N}{N+2}<p\le \frac{2N}{N+1}$
  • Figure 5: The case $\frac{2N}{N+\sigma}<p\le \frac{2N}{N+2}$

Theorems & Definitions (37)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • ...and 27 more