Critical exponent for nonlinear nonlocal equations

Abstract

In this thesis, we study the nonlinear nonlocal equation ∂t u=J*u-u+u¹+p where p>0 and J is nonnegative, bounded, and radially symmetric with unit integral. The Fourier transform of J satisfies J ̂(ξ)=1-A|ξ|^β (ln1/|ξ| 〗 )^μ+o(〖|ξ|〗^β (ln〖1/(|ξ|)〗 )^μ ) as ξ→0, for 0<β≤2, μ∈R, and A>0. In this study, we establish the local existence, uniqueness, and the comparison principle of solutions. Furthermore, we show that the Fujita critical exponent for this equation is β/n when μ<0. For the case μ>1, we discover that the critical exponent is β/n , and the solutions belong to the global to the small initial data regime.