Higher Hölder regularity for nonlocal parabolic equations with irregular kernels. (English) Zbl 1527.35108
In this paper, the authors prove Hölder regularity for the parabolic equation
\begin{align*}
\partial_t u - \mathrm{pv}\int\limits_{\mathbb{R}^n}\Phi(u(x,t)-u(y,t))\frac{A(x,y,t)}{|x-y|^{n+2s}}\,dy=f,
\end{align*}
where \(\Phi:\mathbb{R}\to\mathbb{R}\) satisfies \(\Phi(0)=0\) and standard growth conditions
\begin{align*}
(\Phi(\xi)-\Phi(\xi'))(\xi-\xi')\gtrsim |\xi-\xi'|^2,\\
|\Phi(\xi)-\Phi(\xi')|\lesssim|\xi-\xi'|,
\end{align*}
\(A:\mathbb{R}^n\times\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}\) is measurable function bounded from above and below, which is “locally close enough to being translation invariant”, and \(f\) satisfies a suitable integrability condition. This condition allows for discontinuous kernels.
For \(\Phi(t)=t\) and \(A=1\), the equation is the fractional heat equation.
The condition on \(A\) is a parabolic version of the condition presented for nonlocal elliptic equations in [S. Nowak, Calc. Var. Partial Differ. Equ. 60, No. 1, Paper No. 24, 37 p. (2021; Zbl 1509.35087)].
The proofs use a perturbation argument similar to [S. Nowak, Calc. Var. Partial Differ. Equ. 60, No. 1, Paper No. 24, 37 p. (2021; Zbl 1509.35087)] and the method of iterated discrete differentiation from [L. Brasco et al., J. Evol. Equ. 21, No. 4, 4319–4381 (2021; Zbl 1486.35084)].
Additionally, the authors prove existence and local boundedness for this equation without the extra constraint on \(A\).
For \(\Phi(t)=t\) and \(A=1\), the equation is the fractional heat equation.
The condition on \(A\) is a parabolic version of the condition presented for nonlocal elliptic equations in [S. Nowak, Calc. Var. Partial Differ. Equ. 60, No. 1, Paper No. 24, 37 p. (2021; Zbl 1509.35087)].
The proofs use a perturbation argument similar to [S. Nowak, Calc. Var. Partial Differ. Equ. 60, No. 1, Paper No. 24, 37 p. (2021; Zbl 1509.35087)] and the method of iterated discrete differentiation from [L. Brasco et al., J. Evol. Equ. 21, No. 4, 4319–4381 (2021; Zbl 1486.35084)].
Additionally, the authors prove existence and local boundedness for this equation without the extra constraint on \(A\).
Reviewer: Vivek Tewary (Sri City)
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35D30 | Weak solutions to PDEs |
| 35K10 | Second-order parabolic equations |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 35R09 | Integro-partial differential equations |
| 47G20 | Integro-differential operators |
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