ON THE STABILITY OR INSTABILITY OF THE SINGULAR SOLUTION OF THE SEMILINEAR HEAT-EQUATION WITH EXPONENTIAL REACTION TERM

We study questions of existence, uniqueness and asymptotic behaviour f or the solutions of u(x, t) of the problem u(t) - Delta u = lambda e(u ), lambda > 0, t > 0, x is an element of B. (P) u(x, 0) = u(0)(x), x i s an element of B, u(x, t) = 0 on partial derivative B x (0, infinity) , where B is the unit ball {x is an element of R(N): \x\ less than or equal to 1} and N greater than or equal to 3. Our interest is focused on the parameter lambda(0) = 2(N - 2) for which (P) admits a singular stationary solution of the form S(x)= -2log\x\. We study the dynamical stability or instability of S, which depends on the dimension. In par ticular, there exists a minimal bounded stationary solution u which is stable if 3 less than or equal to N less than or equal to 9, while S is unstable. For N greater than or equal to 10 there is no bounded min imal solution and S is an attractor from below but not from above. In fact, solutions larger than S cannot exist in any time interval (there is instantaneous blow-up), and this happens for all dimensions.