In the first part, we derive sufficient conditions for global existence of solutions of semilinear parabolic problems in a smoothly bounded domain $\Omega\subset R^n$. These conditions require the boundedness of solutions in L_p-spaces. Our examples include some weakly coupled systems, problems involving measures, nonlinear boundary conditions and nonlocal nonlinearities.

In the second part, we use the results mentioned above for the proof of a priori estimates of global solutions of some superlinear parabolic problems and we discuss some applications of these estimates.

[BACK]