The following problem is discussed:
in , , if , ; if , , , on , , ,
As grows with t, it is assumed that is the supremum of all such that the solution to the problem above exists for , and that is a blow-up point if there is , , , and ,
A partial outline of the results can be divided into:
Case (i): is a ball, u(.,t) are radial functions, . The authors prove that the only blow-up point is . For the case , , , they obtain: , any , , , , ; if moreover and ; or and , as provided , some
Case (ii): Non symmetric, is a convex domain: the blow-up points lie in a compact subset of . For the particular , u(x,t), all , and if
In the one-dimensional case , if ’ changes sign just once, then the solution blow-up at a single point. The authors extend some of the results to the case of a boundary condition , , , and the outward normal vector.