Nonexistence of global solutions for a class of nonlocal in time and space nonlinear evolution equations

In this paper, we study the nonlocal nonlinear evolution equation CD0|t αu(t,x)−(J∗|u|−|u|)(t,x)+CD0|t βu(t,x)=|u(t,x)|p,t>0,x∈Rd,where 1<α<2, 0<β<1, p>1, J:Rd→R+, ∗ is the convolution product in Rd, and CD0|t q, q∈{α,β}, is the Caputo left-sided fractional derivative of order q with respect to the time t. We prove that the problem admits no global weak solution other than the trivial one with suitable initial data when 1<p<1+[Formula presented]. Next, we deal with the system CD0|t αu(t,x)−(J∗|u|−|u|)(t,x)+CD0|t βu(t,x)=|v(t,x)|p,t>0,x∈Rd,CD0|t αv(t,x)−(J∗|v|−|v|)(t,x)+CD0|t βv(t,x)=|u(t,x)|q,t>0,x∈Rd,where 1<α<2, 0<β<1, p>1, and q>1. We prove that the system admitsnon global weak solution other than the trivial one with suitable initial data when 1<pq<1+[Formula presented]max{p+1,q+1}. Our approach is based on the test function method. © 2018 Elsevier Ltd

Авторы
Jleli M.1 , Kirane M. 2, 3, 4 , Samet B. 1
Редакторы
-
Издательство
Elsevier Ltd
Номер выпуска
8
Язык
Английский
Страницы
2698-2709
Статус
Опубликовано
Подразделение
-
Номер
-
Том
75
Год
2018
Организации
  • 1 Department of Mathematics, College of Science, King Saud University, Riyadh, 11451, Saudi Arabia
  • 2 LaSIE, Faculté des Sciences et Technologies, Université de La Rochelle, Avenue M. Crépeau, La Rochelle, Cedex, 17042, France
  • 3 RUDN University, 6 Miklukho-Maklay St, Moscow, 117198, Russian Federation
  • 4 Nonlinear Analysis and Applied Mathematics, (NAAM) Research Group Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
Ключевые слова
Caputo fractional derivative; Global solution; Nonexistence
Дата создания
19.10.2018
Дата изменения
19.10.2018
Постоянная ссылка
https://repository.rudn.ru/ru/records/article/record/6715/