Qualitative properties of solutions to a generalized Fisher-KPP equation

The following Fisher-KPP type equation $$u_t=K{u}_{xx}-B{u}^q+A{u}^p,(x,t)\in \mathrm{\Re }\times (0,\mathrm{\infty }),$$ with $p>q>0$ and $A$, $B$, $K$ positive coefficients, is considered. For both $p>q>1$ and $p>1$, $q=1$, we construct stationary solutions, establish their behavior as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ and prove that they are separatrices between solutions decreasing to zero in infinite time and solutions presenting blow-up in finite time. We also establish decay rates for the solutions that decay to zero as $t\to \mathrm{\infty }$.