An $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients
By Ildoo Kim, Kyeong-Hun Kim, Sungbin Lim. 2015
75 citations recorded by the source corpus.
We introduce an $L_q(L_p)$-theory for the quasi-linear fractional equations of the type $$ \partial^{\alpha}_t u(t,x)=a^{ij}(t,x)u_{x^i x^j}(t,x)+f(t,x,u), \quad t>0, \,x\in \mathbf{R}^d. $$ Here, $\alpha\in (0,2)$, $p,q>1$, and $\partial^{\alpha}_t$ is the Caupto fractional derivative of order $\alpha$. Uniqueness, existence, and $L_q(L_p)$-estimates of solutions are obtained. The leading coefficients $a^{ij}(t,x)$ are assumed to be piecewise continuous in $t$ and uniformly continuous in $x$. In particular $a^{ij}(t,x)$ are allowed to be discontinuous with respect to the time variable. Our approach is based on classical tools in PDE theories such as the Marcinkiewicz interpolation theorem, the Calderon-Zygmund theorem, and perturbation arguments.
Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/8534b6dfe7cbd79ff28951dcca1a0481035e265e
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In Applied Mathematics
In Modeling and Simulation
In Numerical Analysis
Citations in the corpus
- L-estimates for time fractional parabolic equations with coefficients measurable in time (2018, methodology): “In a recent paper [6] the authors proved the unique solvability of solutions in mixed Lp,q spaces to the time fractional parabolic equation (1.”
- Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview (2018, background): “, time-dependent diffusion coefficients and Fokker-Planck equation, energy arguments [99] or perturbation arguments [54] can be used to show existence and uniqueness of the solution.”