Blow-up conditions of nonlinear parabolic equations and systems under mixed nonlinear boundary conditions

In this paper, we firstly discuss blow-up phenomena for nonlinear parabolic equations u(t) = del.[rho(u)del u] + f(x, t, u), in Omega x (0, t*), under mixed nonlinear boundary conditions partial derivative u/partial derivative n + theta(z)u = h(z, t, u) on Gamma(1) x (0, t*) and u = 0 on Gamma(2) x (0, t*), where Omega is a bounded domain and Gamma(1) and Gamma(2) are disjoint subsets of a boundary a Q. Here, f and h are real-valued C-1-functions and rho is a positive C-1-function. To obtain the blow-up solutions, we introduce the following blow-up conditions: (2 + is an element of) integral(u)(0) rho(w)f(x, t, w)dw <= u rho(u)f(x, t, u) + beta(1)u(2) + gamma(1), (C rho) : (2 + is an element of) integral(u)(0) rho(2)(w)h(z, t, w)dw <= u rho(2)(u)h(z, t, u) + beta(2)u(2) + gamma(2), for x is an element of Omega, z is an element of partial derivative Omega, t > 0, and u is an element of R for some constants is an element of, and beta(1), beta(2), gamma(1), and gamma(2) satisfying is an element of > 0, beta(1) + lambda(R) + 1/lambda S beta 2 <= rho(2)(m)lambda(R)/2 is an element of and 0 <= beta(2) <= rho(2)(m)lambda(S)/2 is an element of, where rho(m) := inf(s>0) rho(S), lambda(R) is the first Robin eigenvalue and lambda(S) is the first Steklov eigenvalue. Lastly, we discuss blow-up solutions for nonlinear parabolic systems.