Green’s Function Estimates for Time-Fractional Evolution Equations
| Autor: | Vassili N. Kolokoltsov, Ifan Johnston |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Statistics and Probability
Constant coefficients lcsh:Analysis Green’s function lcsh:Thermodynamics 01 natural sciences Caputo derivative 010104 statistics & probability symbols.namesake Mathematics - Analysis of PDEs lcsh:QC310.15-319 FOS: Mathematics fractional evolution Order (group theory) 0101 mathematics Mathematics Variable (mathematics) Kernel (set theory) Operator (physics) lcsh:Mathematics 010102 general mathematics Mathematical analysis Probability (math.PR) lcsh:QA299.6-433 Statistical and Nonlinear Physics Function (mathematics) lcsh:QA1-939 Elliptic operator TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES Green's function 60J35 (Primary) 60G52 35J08 33E12 44A10 60J75 (Secondary) symbols Aronson estimates two-sided estimates Analysis Mathematics - Probability Analysis of PDEs (math.AP) |
| Zdroj: | Fractal and Fractional, Vol 3, Iss 2, p 36 (2019) Fractal and Fractional Volume 3 Issue 2 |
| ISSN: | 2504-3110 |
| Popis: | We look at estimates for the Green&rsquo s function of time-fractional evolution equations of the form D 0 + * &nu u = L u , where D 0 + * &nu is a Caputo-type time-fractional derivative, depending on a Lé vy kernel &nu with variable coefficients, which is comparable to y - 1 - &beta for &beta &isin ( 0 , 1 ) , and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green&rsquo s function of D 0 &beta u = L u in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green&rsquo u = &Psi ( - i &nabla ) u where &Psi is a pseudo-differential operator with constant coefficients that is homogeneous of order &alpha Thirdly, we obtain local two-sided estimates for the Green&rsquo u = L u where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green&rsquo s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green&rsquo s functions associated with L and &Psi as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D 0 ( &nu t ) u = L u , where D ( &nu t ) is a Caputo-type operator with variable coefficients. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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