{"title":"Application of capacities to space–time fractional dissipative equations I: regularity and the blow-up set","authors":"Pengtao Li, Zhichun Zhai","doi":"10.4153/s0008414x22000566","DOIUrl":null,"url":null,"abstract":"Abstract We apply capacities to explore the space–time fractional dissipative equation: (0.1) \n$$ \\begin{align} \\left\\{\\begin{aligned} &\\partial^{\\beta}_{t}u(t,x)=-\\nu(-\\Delta)^{\\alpha/2}u(t,x)+f(t,x),\\quad (t,x)\\in\\mathbb R^{1+n}_{+},\\\\ &u(0,x)=\\varphi(x),\\ x\\in\\mathbb R^{n}, \\end{aligned}\\right. \\end{align} $$\n where \n$\\alpha>n$\n and \n$\\beta \\in (0,1)$\n . In this paper, we focus on the regularity and the blow-up set of mild solutions to (0.1). First, we establish the Strichartz-type estimates for the homogeneous term \n$R_{\\alpha ,\\beta }(\\varphi )$\n and inhomogeneous term \n$G_{\\alpha ,\\beta }(g)$\n , respectively. Second, we obtain some space–time estimates for \n$G_{\\alpha ,\\beta }(g).$\n Based on these estimates, we prove that the continuity of \n$R_{\\alpha ,\\beta }(\\varphi )(t,x)$\n and the Hölder continuity of \n$G_{\\alpha ,\\beta }(g)(t,x)$\n on \n$\\mathbb {R}^{1+n}_+,$\n which implies a Moser–Trudinger-type estimate for \n$G_{\\alpha ,\\beta }.$\n Then, for a newly introduced \n$L^{q}_{t}L^p_{x}$\n -capacity related to the space–time fractional dissipative operator \n$\\partial ^{\\beta }_{t}+(-\\Delta )^{\\alpha /2},$\n we perform the geometric-measure-theoretic analysis and establish its basic properties. Especially, we estimate the capacity of fractional parabolic balls in \n$\\mathbb {R}^{1+n}_+$\n by using the Strichartz estimates and the Moser–Trudinger-type estimate for \n$G_{\\alpha ,\\beta }.$\n A strong-type estimate of the \n$L^{q}_{t}L^p_{x}$\n -capacity and an embedding of Lorentz spaces are also derived. Based on these results, especially the Strichartz-type estimates and the \n$L^{q}_{t}L^p_{x}$\n -capacity of fractional parabolic balls, we deduce the size, i.e., the Hausdorff dimension, of the blow-up set of solutions to (0.1).","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"299 8","pages":"1904 - 1956"},"PeriodicalIF":0.6000,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4153/s0008414x22000566","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract We apply capacities to explore the space–time fractional dissipative equation: (0.1)
$$ \begin{align} \left\{\begin{aligned} &\partial^{\beta}_{t}u(t,x)=-\nu(-\Delta)^{\alpha/2}u(t,x)+f(t,x),\quad (t,x)\in\mathbb R^{1+n}_{+},\\ &u(0,x)=\varphi(x),\ x\in\mathbb R^{n}, \end{aligned}\right. \end{align} $$
where
$\alpha>n$
and
$\beta \in (0,1)$
. In this paper, we focus on the regularity and the blow-up set of mild solutions to (0.1). First, we establish the Strichartz-type estimates for the homogeneous term
$R_{\alpha ,\beta }(\varphi )$
and inhomogeneous term
$G_{\alpha ,\beta }(g)$
, respectively. Second, we obtain some space–time estimates for
$G_{\alpha ,\beta }(g).$
Based on these estimates, we prove that the continuity of
$R_{\alpha ,\beta }(\varphi )(t,x)$
and the Hölder continuity of
$G_{\alpha ,\beta }(g)(t,x)$
on
$\mathbb {R}^{1+n}_+,$
which implies a Moser–Trudinger-type estimate for
$G_{\alpha ,\beta }.$
Then, for a newly introduced
$L^{q}_{t}L^p_{x}$
-capacity related to the space–time fractional dissipative operator
$\partial ^{\beta }_{t}+(-\Delta )^{\alpha /2},$
we perform the geometric-measure-theoretic analysis and establish its basic properties. Especially, we estimate the capacity of fractional parabolic balls in
$\mathbb {R}^{1+n}_+$
by using the Strichartz estimates and the Moser–Trudinger-type estimate for
$G_{\alpha ,\beta }.$
A strong-type estimate of the
$L^{q}_{t}L^p_{x}$
-capacity and an embedding of Lorentz spaces are also derived. Based on these results, especially the Strichartz-type estimates and the
$L^{q}_{t}L^p_{x}$
-capacity of fractional parabolic balls, we deduce the size, i.e., the Hausdorff dimension, of the blow-up set of solutions to (0.1).
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