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Existence results for a Cauchy-Dirichlet parabolic problem with a repulsive gradient term

Martina Magliocca

TL;DR

This work analyzes existence for a nonlinear parabolic Cauchy-Dirichlet problem with a gradient-dependent lower-order term H(t,x,∇u) that grows like γ|∇u|^q plus a forcing f. By exploiting Leray–Lions type operators and a careful distinction between sublinear, linear, and superlinear gradient growth, the authors establish finite-energy and renormalized-solutions theories under sharp relations between q, p, N, and the data spaces of the initial datum and forcing. The paper derives extensive a priori estimates, compactness, and convergence results for approximating problems, enabling existence results across multiple regimes (finite-energy, infinite-energy, and L^1 data). It thus extends parabolic gradient-growth analysis to general divergence-form operators with measurable coefficients, bridging gaps between sublinear growth and natural-growth settings. The results have implications for well-posedness of parabolic PDEs with gradient nonlinearities in heterogeneous media and contribute a robust framework for handling unbounded data and gradient-driven source terms.

Abstract

We study the existence of solutions of a nonlinear parabolic problem of Cauchy-Dirichlet type having a lower order term which depends on the gradient. The model we have in mind is the following: \[ \begin{cases}\begin{split} & u_t-\text{div}(A(t,x)\nabla u|\nabla u|^{p-2})=γ|\nabla u|^q+f(t,x) &\qquad\text{in } Q_T,\\ & u=0 &\qquad\text{on }(0,T)\times \partial Ω,\\ & u(0,x)=u_0(x) &\qquad\text{in } Ω, \end{split}\end{cases} \] where $Q_T=(0,T)\times Ω$, $Ω$ is a bounded domain of $\mathrm{R}^N$, $N\ge 2$, $1<p<N$, the matrix $A(t,x)$ is coercive and with measurable bounded coefficients, the r.h.s. growth rate satisfies the superlinearity condition \[ \max\left\{\frac{p}{2},\frac{p(N+1)-N}{N+2}\right\}<q<p \] and the initial datum $u_0$ is an unbounded function belonging to a suitable Lebesgue space $L^σ(Ω)$. We point out that, once we have fixed $q$, there exists a link between this growth rate and exponent $σ=σ(q,N,p)$ which allows one to have (or not) an existence result. Moreover, the value of $q$ deeply influences the notion of solution we can ask for. The sublinear growth case with \[ 0<q\le\frac{p}{2} \] is dealt at the end of the paper for what concerns small value of $p$, namely $1<p<2$.

Existence results for a Cauchy-Dirichlet parabolic problem with a repulsive gradient term

TL;DR

This work analyzes existence for a nonlinear parabolic Cauchy-Dirichlet problem with a gradient-dependent lower-order term H(t,x,∇u) that grows like γ|∇u|^q plus a forcing f. By exploiting Leray–Lions type operators and a careful distinction between sublinear, linear, and superlinear gradient growth, the authors establish finite-energy and renormalized-solutions theories under sharp relations between q, p, N, and the data spaces of the initial datum and forcing. The paper derives extensive a priori estimates, compactness, and convergence results for approximating problems, enabling existence results across multiple regimes (finite-energy, infinite-energy, and L^1 data). It thus extends parabolic gradient-growth analysis to general divergence-form operators with measurable coefficients, bridging gaps between sublinear growth and natural-growth settings. The results have implications for well-posedness of parabolic PDEs with gradient nonlinearities in heterogeneous media and contribute a robust framework for handling unbounded data and gradient-driven source terms.

Abstract

We study the existence of solutions of a nonlinear parabolic problem of Cauchy-Dirichlet type having a lower order term which depends on the gradient. The model we have in mind is the following: where , is a bounded domain of , , , the matrix is coercive and with measurable bounded coefficients, the r.h.s. growth rate satisfies the superlinearity condition and the initial datum is an unbounded function belonging to a suitable Lebesgue space . We point out that, once we have fixed , there exists a link between this growth rate and exponent which allows one to have (or not) an existence result. Moreover, the value of deeply influences the notion of solution we can ask for. The sublinear growth case with is dealt at the end of the paper for what concerns small value of , namely .

Paper Structure

This paper contains 25 sections, 22 theorems, 260 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Comments on the $q$ growth and comparison with the stationary problem
  3. On the superlinear setting
  4. Assumptions and statements
  5. Assumptions
  6. Statements of the main results and comments
  7. Plan of the paper
  8. Solutions of finite energy
  9. The a priori estimate
  10. The a.e. convergence of the gradient
  11. The existence result
  12. Solutions of infinite energy
  13. The a priori estimate
  14. Some convergence results
  15. The existence result
  16. ...and 10 more sections

Key Result

Theorem 1

Let $1<p<N$ and assume A1, A2, A3, ID1, F1 and H with either or Then, there exists at least one finite energy solution of the problem pb (see Definition sol). Moreover, this solution fulfils the following regularities: and

Figures (6)

  • 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 $1<p\le \frac{2N}{N+2}$
  • ...and 1 more figures

Theorems & Definitions (40)

  • Remark 3.1
  • Theorem 1: Red zone
  • Theorem 2: Orange zone
  • Theorem 3: Yellow zone
  • Definition 4.1
  • Theorem 4.2
  • Corollary 4.3
  • Proposition 4.4
  • Theorem 4.5
  • Definition 5.1
  • ...and 30 more