Global nonexistence for the Cauchy problem of some nonlinear reaction-diffusion systems
This paper is devoted to a series of problems concerning critical exponents for some nonlinear reaction-diffusion equations. They start by considering the nonautonomous diffusive-convective parabolic equation with a fractal diffusion \[ u_t=- (-\Delta)^{\alpha/2} u^m+a(x)\cdot\nabla u^q+f (t,x)|u|^p+ w(t,x), \tag{\(E_m\)} \] for \(t>0\), \(x\in\mathbb{R}^N\), supplemented with the initial condition \(u(0,x)=u_0(x)\), \(x\in\mathbb{R}^N\), where \(m,p,\) and \(q\) are positive reals. The authors find conditions relating the space dimension \(N\) with the parameters \(\alpha,m,q\), and \(p\) for which every global nontrivial solution of \((E_m)\) is trivial. The results of the authors answer positively some open problems raised recently by Bandle, Dengs, Levine, and Zhang.
scientific article; zbMATH DE number 1763454
scientific article