Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4958519 | Computers & Mathematics with Applications | 2017 | 13 Pages |
Abstract
In this paper, we consider the fractional heat equation ut=â³Î±/2u+f(u) with Dirichlet conditions on the ball BRâRd, where â³Î±/2 is the fractional Laplacian, f:[0,â)â[0,â) is continuous and non-decreasing. We present the characterisations of f to ensure the equation has a local solution in Lq(BR) provided that the non-negative initial data u0âLq(BR). For q>1 and 1<α<2, we show that the equation has a local solution in Lq(BR) if and only if limsââsupsâ(1+αq/d)f(s)=â; and for q=1 and 1<α<2 if and only if â«1âsâ(1+α/d)F(s)ds<â, where F(s)=sup1â¤tâ¤sf(t)/t. When limsâ0f(s)/s<â, the same characterisations holds for the fractional heat equation on the whole space Rd.
Related Topics
Physical Sciences and Engineering
Computer Science
Computer Science (General)
Authors
Kexue Li,