A nonlocal in time parabolic system whose Fujita critical exponent is not given by scaling

Abstract

We consider the nonlocal coupled parabolic systemary. We assume that 0 γ 1 , γ 2 < 1, p, q 1 and the initial data u(0), v(0) ∈ C 0 (Ω). We obtain the Fujita critical exponent for the system above which is not given by the scaling argument.

Existence and blow-up of solutions for damped ave system with nonlinear memory

“…Motivated by the papers [10,11,14,15], in the present paper, we consider the problem (1), we will give conditions relating the space dimension N with the system of parameters γ 1 ;γ 2 ; p and q for which the solution exists globally in time and satisfies L ∞ -decay estimates, as well as a blow-up in finite time of solution with initial data having positive average.…”

“…In order to do this, we are going to use L p − L q estimates of the fundamental solutions for linear damped wave equation derived by Nishihara [6,7], Hosono and Ogawa [16], combined with Escobedo and Herrero's [17] technique, which have been recently used to study problems of the parabolic and hyperbolic systems type, (see for instance [15,18,19]) and the references therein.…”

Global Small data Solutions for a system of semilinear heat equations and the corresponding system of damped wave equations with nonlinear memory

“…(i) If γ = 0, p ≤ p * , and u 0 ≥ 0, u 0 ≡ 0, then u blows up in finite time. This result is later extended by some authors to the weakly coupled parabolic systems, damped wave equations, weakly coupled damped wave systems, we refer the reader to [22,42,39,5,4,40,41,1,44]. We just briefly describe the results directly connected to our problems.…”

“…On the other hand, it seems that there are a few results related to the problems (1.1) with time nonlocal nonlinear source as compared with nonlinear polynomial source (see for instance ([3], [22], [1], [42], [39], [40], [4], [5], [41]). Here we only mention some of them motivate our work.…”

Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities

“…A first change of critical exponent thus occurs when q decreases from n+1 n , but in a continuous way, and a discontinuous jump of the critical exponent then occurs when q reaches 1 from above. We refer to [12,24] for general survey articles on critical Fujita exponents and to [10,27,13] for other non-standard behaviors regarding such exponents (appearing in certain time-nonlocal problems).…”