Remarks on the Cauchy problem for the one-dimensional quadratic (fractional) heat equation

被引:5
作者
Molinet, Luc [1 ]
Tayachi, Slim [2 ]
机构
[1] Univ Tours, Lab Math & Phys Theor, CNRS UMR 7350, Federat Denis Poisson, F-37200 Tours, France
[2] Univ Tunis, Dept Math, Fac Sci Tunis, Tunis 2092, Tunisia
关键词
Nonlinear heat equation; Fractional heat equation; Ill-posedness; Well-posedness; SIMILAR GLOBAL-SOLUTIONS; ILL-POSEDNESS; WELL-POSEDNESS; PARABOLIC EQUATIONS; EXISTENCE; NONEXISTENCE;
D O I
10.1016/j.jfa.2015.08.002
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We prove that the Cauchy problem associated with the one dimensional quadratic (fractional) heat equation: u(t) = -(-Delta)(alpha)u -/+ u(2), t is an element of(0,T), x is an element of R or T, with 0 < alpha <= 1 is well-posed in H-s for s >= max(-alpha, 1/2 - 2 alpha) except in the case alpha = 1/2 where it is shown to be well-posed for s > -1/2 and ill-posed for s = -1/2. As a by-product we improve the known well-posedness results for the heat equation (alpha = 1) by reaching the end-point Sobolev index s = -1. Finally, in the case 1/2 < alpha <= 1, we also prove optimal results in the Besov spaces B-2(s,q). (C) 2015 Elsevier Inc. All rights reserved.
引用
收藏
页码:2305 / 2327
页数:23
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