Blow-up of solutions of semilinear parabolic equations
Consider the blow-up behavior of solutions of a semilinear parabolic equation \[ (1)\quad u_ t+uu_ x-u_{xx}=f(u)\quad (-a<x<a,\quad t>0) \] with the initial-boundary conditions \(u(\pm a,t)=0\) \((t>0)\), \(u(x,0)=\phi (x)\) \((-a<x<a)\), where \(f(u)=u^ p\) if \(u\geq 0\), \(p>2\), \(\phi (\pm a)=0\), \(\phi\geq 0\) \((-a<x<a).\) This paper shows that (i) a finite time blow-up occurs if \(\phi\) is large enough; (ii) there is a single point blow-up if \(p>3\) and for a large class of initial values \(\phi\) ; (iii) if \(p>3\) then for a large class of \(\phi\) 's \[ u(x,t)\leq c(T-t)^{-1/(p-1)}\quad (c>0). \] These results can be extended to equations \(u_ t-u_{xx}=F(u,u_ x)\).
scientific article; zbMATH DE number 4101717
scientific article