Some Blow-up Problems For A Semilinear Heat Equation With A Potential

In this thesis,we study a semilinear parabolic equation with potential where ?(?)RN(N?3)is a bounded,smooth domain,initial datum u0 ? L?(?),p>1 and V ? C1(?)with positive lower bound.We prove that when 1<p<ps=n+2/n-2,the blow-up solutions to the problem must be completely blow-up solutions.In critical case,all finite time non-collapsing blow-up must be refined second type blow-up.Secondly,we show that the Hausdorff dimensions of the blow-up set S of the energy non-collapsing maximal solution,does not exceed N-2-4/p-1.Finally,examples of complete blow-up and incomplete blow-up are given.Due to the appearance of the potential function,the energy derivative is not mono-tone decreasing anymore.Notice that the integral ??s|(?)V/(?)s||w|p+1 ?dy can be controlled by e-s/2 ??s|y|w|p+1?dy.Introducing a new energery function,we can get the quasi-montonicity formula and obtain some useful estimates of the solutions.