Second critical exponent and life span for pseudo-parabolic equation (Q1759898)

The authors study the second critical exponent and the life span for the equation \[ u_t-k\Delta u_t=\Delta u+u^p\quad (x,t)\in{\mathbb R}^n\times(0,T) \] with initial condition \(u(x,0)=u_0(x)\) and parameters \(p>1\) and \(k>0\). They consider the problem in an equivalent form as nonlocal diffusion equation instead of the standard energy estimates method. It is proved that the decay order of the initial data required by global solutions in the coexisting region of global and non-global solutions depends on the pseudo-parabolic parameter \(k\). In the last part, the authors give some numerical examples to illustrate all obtained results.