BLOW-UP AT SPACE INFINITY FOR NONLINEAR HEAT EQUATIONS

Abstract

The last condition of (A1) forces f to grow superlinearly at infinity. A nonlinear evolution equation may have a unique local-in-time solution in a suitable function space and it can be extend as a solution together with evolution of time so long as it belongs to the function space. However, in general, the Cauchy problem is not solvable globally in time; a solution may blow up in finite time. That is, there may exist a finite time T < ∞ such that the solution ceases to exist in the function space at the time T . This phenomenon is called blow-up in finite time and we call such a time T blow-up time. The Cauchy problem (1.1) has a unique local-in-time solution u = u(·, t) in L∞(RN ) for any nonnegative initial data u0 ∈ L∞(RN ). However, it may blow up in finite time. For instance, if the initial data does not decrease at space infinity, the solution of (1.1) does blow up in finite time. We are interested in the blow-up times of solutions and detailed behavior of solutions at the blow-up times. In particular, we discuss solutions which blow up at space infinity as we will state later.