Abstract
We study the existence and nonexistence of global weak solutions to the semilinear parabolic differential inequality
partial derivative(t)u - Delta u >= vertical bar u vertical bar(p), (t, x) is an element of (0, infinity) x B-c,
where p > 1, B is the closed unit ball in R-N (N >= 2) and B-c is its complement, under the semilinear dynamical boundary conditions
partial derivative(t)u + u >= vertical bar u vertical bar(q) + w(x), (t, x) is an element of (0, infinity) x partial derivative B
or
partial derivative(t)u + partial derivative(nu)u + alpha u >= vertical bar u vertical bar(q) + w(x), (t, x) is an element of (0, infinity) x partial derivative B,
where q > 1, alpha >= 0, partial derivative(nu) := partial derivative/partial derivative nu+, nu(+) is the outward unit normal (relative to B-c) on partial derivative B and w is an element of L-1(partial derivative B), integral(partial derivative B) w(x) dS(x) >= 0. The cases integral(partial derivative B) w(x) dS(x) = 0 and integral(partial derivative B) w(x) dS(x) > 0 are discussed separately.