Linear and nonlinear heat equations in L-delta(q) spaces and universal bounds for global solutions

We develop a theory of both linear and nonlinear heat equations in the weig hted Lebesgue spaces L-delta(q) where delta is the distance to the boundary . In particular, we prove an optimal L-delta(q) - L-delta(r) estimate for t he heat semigroup, and we establish sharp results on local existence-unique ness and local nonexistence of solutions for semilinear heat equations with initial values in those spaces. This theory enables us to obtain new types of results concerning positive global solutions of superlinear parabolic p roblems. Namely, under certain assumptions, we prove that any global soluti on is uniformly bounded for t less than or equal to tau > 0 by a universal constant, independent of the initial data. In all previous results, the bou nds for global solutions were depending on the initial data.