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Parabolic problems with slightly superlinear convection terms

Fessel Achhoud

TL;DR

This work analyzes a nonlinear parabolic PDE with a slightly superlinear logarithmic convection term given by $u_t - \mathrm{div}(\mathcal{M}(x,t)\nabla u) = -\mathrm{div}(h(u)E) + f$, where $h(l)=l\log(e+|l|)$. The authors establish both bounded and unbounded weak solutions under precise integrability conditions on the data, using truncation approximations, energy estimates, and a robust comparison principle. A key contribution is the development of level-set decay estimates and compactness arguments that yield existence and uniqueness in energy spaces, with extensions to more general growth and convection structures. The results extend classical Boccardo-type theory to parabolic problems with logarithmic, slightly superlinear convection terms, providing a rigorous framework for noncoercive operators in this setting.

Abstract

In this paper we deal with a non-linear parabolic problem which involving a convection term with super--linear growth, whose model is \[ \frac{\partial u}{\partial t}-÷(\mathcal{M}(x,t)\nabla u)= -÷(u\log (e+|u|)E(x,t))+f(x,t), \] where $\mathcal{M}$ is a bounded measurable matrix, the vector field $E$ and the function $f$ belong to suitable Lebesgue spaces. We prove the existence of a unique bounded and unbounded weak solution.

Parabolic problems with slightly superlinear convection terms

TL;DR

This work analyzes a nonlinear parabolic PDE with a slightly superlinear logarithmic convection term given by , where . The authors establish both bounded and unbounded weak solutions under precise integrability conditions on the data, using truncation approximations, energy estimates, and a robust comparison principle. A key contribution is the development of level-set decay estimates and compactness arguments that yield existence and uniqueness in energy spaces, with extensions to more general growth and convection structures. The results extend classical Boccardo-type theory to parabolic problems with logarithmic, slightly superlinear convection terms, providing a rigorous framework for noncoercive operators in this setting.

Abstract

In this paper we deal with a non-linear parabolic problem which involving a convection term with super--linear growth, whose model is where is a bounded measurable matrix, the vector field and the function belong to suitable Lebesgue spaces. We prove the existence of a unique bounded and unbounded weak solution.

Paper Structure

This paper contains 3 sections, 7 theorems, 87 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminary results
  3. Proof of the existence results

Key Result

Theorem 1.1

Assume alfa, ipoh and take $E\in [L^{r}(\mathcal{Q})]^N$, with $r>N+2$, and $f\in L^{m}(\mathcal{Q})$, with $m>\frac{N}{2}+1$. Then, there exists a unique weak solution $u\in L^2(0,T; W^{1,2}_0(\Omega))\cap L^{\infty}(\mathcal{Q})$ to problem problem.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 9 more