Critical behavior of a semilinear time fractional diffusion equation with forcing term depending on time and space

Abstract

In this paper we study the time fractional semilinear diffusion equation 0CDtαu(t,x)−Δu(t,x)=|u|p+tσw(x) with the initial conditions u(0,x)=u0(x) and ∂tu(0,x)=u1(x) for x∈RN, where α∈(0,1)∪(1,2), p>1, −1<σ<0 and w⁄≡0, u1=0 when 0<α<1. The novelty of this paper lies in considering semilinear time-fractional diffusion equations with a forcing term tσw(x) which depends on time and space. We show that there are critical exponents in the following cases. (i) For α+σ>0, the solution of the above subdiffusion equation blows up in finite time when 1<p<1+2ααN−2σ−2α and ∫RNw(x)dx>0, while the global solution exists for suitably small initial data u0 and w belonging to certain Lebesgue spaces when p≥1+2ααN−2σ−2α. (ii) The solution of the above superdiffusion equation blows up in finite time when 1<p≤1+2ααN−2σ−2α and ∫RNw(x)dx>0, while the global solution exists for suitably small initial data u0 and w belonging to certain Lebesgue spaces when p>1+2ααN−2σ−2α and u1=0. (iii) For α+σ≥1, the solution of the above superdiffusion equation blows up in finite time when 1<p≤1+2ααN−2σ−2α and ∫RNw(x)dx>0, while the global solution exists for suitably small initial data u0, u1 and w belonging to certain Lebesgue spaces when p>1+2ααN−2σ−2α. (iv) For α+σ<1, the solution of the above superdiffusion equation blows up in finite time when 1<p<1+2ααN−2 and ∫RNw(x)dx>0, while the global solution exists for suitably small initial data u0, u1 and w belonging to certain Lebesgue spaces when p>1+2ααN−2. The critical exponent in (iv) is different from that in (iii) and (ii). This peculiarity is related to the fact the time order of the equation and the inhomogeneous are both fractional, and so the role played by the second data u1 becomes “unnatural” as α∈(1,2). Namely, the change of the critical exponent in (iv) is due to that α+σ<1 and u1⁄≡0.