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Regularizing effect for some parabolic problems with perturbed terms and irregular data

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Abstract

In this paper we deal with regularizing effect for some parabolic problems with lower order terms defined through a composition with a continuous, but unbounded, function on some intervals \([0,\sigma )\) with \(\sigma >0\), that have at most a power growth with respect to the gradient and a suitable data. A nonexistence result for some measures as data is also investigated.

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Notes

  1. More precisely, the necessary and sufficient conditions for a set to be removable. For example, if A is a compact subset of \({\mathbb {R}}^{\mathbb{N}}\), then A is removable for \(L^{\infty }({\mathbb {R}}^{\mathbb{N}})\) and the Laplacian \(\Delta \) if and only if the capacity of A is zero; or, if A is a compact subset of C then A is removable for \(L^{\infty }(C)\) if and only if the analytic capacity of A is zero.

  2. This is not difficult to do but still needs few technicalities by assuming \(p>2-\frac{1}{N}\) which is equivalent to \(\frac{N(p-1)}{N-1}>1\).

  3. Note that the equivalence between renormalized and entropy solutions is immediate for such measures.

  4. I.e., \(g(\cdot ,s,\zeta )\) is measurable in \(\Omega \) for any \((s,\zeta )\) in \(\mathbb {R}\times \mathbb {R}^{N}\), and \(g(x,\cdot ,\cdot )\) is continuous in \(\mathbb {R}\times \mathbb {R}^{N}\) for almost every \(x\in \Omega \).

  5. Also called the absolutely continuous measure with respect to the p-capacity, i.e., for every Borel set \(B\subset Q\) such that \(\text {cap}_{p}(B)=0\) it results \(\mu _{d}(B)=0\).

  6. See Notations.

  7. \(a(\cdot ,\cdot ,s,\zeta )\) is measurable on Q for every \((s,\zeta )\) in \(\mathbb {R}\times \mathbb {R}^{N}\) and \(a(t,x,\cdot,\cdot)\) is continuous on \(\mathbb {R}\times \mathbb {R}^{N}\) for a.e. (t, x) in Q.

  8. Let us stress that the assumption \(\gamma (0)=0\) is only technical and it can be removed with the use of a slightly different approximation procedure.

  9. \(M(t,x,s)=(a_{ij}(t,x,s))_{i,j=1}^{N}\) is a symmetric matrix whose coefficients \((a_{i,j}):(0,T)\times \Omega \times \mathbb {R}\mapsto \mathbb {R}\) are Carathéodory functions (i.e., \(a_{i,j}(\cdot ,\cdot ,s)\) is measurable on Q for every \(s\in \mathbb {R}\), and \(a_{i,j}(t,x,\cdot )\) is continuous on \(\mathbb {R}\) for a.e. \((t,x)\in (0,T)\times \Omega \)).

  10. A sequence of functions satisfying

    $$\begin{aligned} \varrho _{n}\in C^{\infty }_{c}(\mathbb {R}^{N+1}),\ \text {supp}(\varrho _{n})\subset \mathcal {B}_{\frac{1}{n}}(0)\text { with } \varrho _{n}\ge 0\text { and }\int _{\mathbb {R}^{N+1}}\varrho _{n}=1. \end{aligned}$$

  11. A “specific” type of positive bump \(C^{\infty }_{c}(Q)\)-functions \(\psi _{\delta }\) (depending on \(\delta >0\)).

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  1. LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, B.P. 1796, Atlas Fez, Morocco

    M. Abdellaoui

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Abdellaoui, M. Regularizing effect for some parabolic problems with perturbed terms and irregular data. J Elliptic Parabol Equ 8, 443–468 (2022). https://doi.org/10.1007/s41808-022-00158-9

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