Asymptotic behavior at the boundary of solutions of reaction-difiusion equations with nonlinear boundary conditions
Claudianor O. Alves · 2005
@u @n = g(x;u); @› in a bounded smooth domain ›. We assume that the nonlinearity f is dissipative (for instance f(x;u) = ifl(x)u p , with fl(x) ‚ 0) while g is explosive (g(x;u) = u q ) and analyze how this two mechanisms compete. We flnd appropriate balances between f and g that will show that the solution starting at any smooth initial condition u0 is bounded for all time. We will show, see (1), that if these balances hold locally around certain point in the boundary, the solution is globally bounded around this point of the boundary. This result complements another one obtained in (2) in which we showed that if the balances between f and g are the opposite then blow-up occurs at that point of the boundary. References