We consider the semilinear Petrovsky equationin a bounded domain and prove the existence of weak solutions. Furthermore, we show that there are solutions under some conditions on initial data which blow up in finite time with non-positive initial energy as well as positive initial energy. Estimates of the lifespan of solutions are also given. Mathematics Subject Classification (2000): 35L35; 35L75; 37B25.

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Communicated by S. ChenThis paper is concerned with global in time behavior of solutions for a quasilinear Petrovsky inverse source problem with boundary dissipation. We establish a stability result when the integral constraint vanishes as time goes to infinity. We also show that the smooth solutions blow up when the data is chosen appropriately.(1.5) has been considered by Amroun and Benaissa in [1], where the authors proved: global existence of solutions by means of the stable set method in H 2 0 . / combined with the Faedo-Galerkin procedure. In [5], Messaoudi studied problem (1.5) when h.u t / D aju t j m 2 u t and proved the existence of a local weak solution and showed that this solution blows up in finite time with negative initial energy if p > m.

In this paper, we study the initial-boundary value problem for a system of nonlinear viscoelastic Petrovsky equations. Introducing suitable perturbed energy functionals and using the potential well method we prove uniform decay of solution energy under some restrictions on the initial data and the relaxation functions. Moreover, we establish a growth result for certain solutions with positive initial energy.

Abstract. Phragmén-Lindelöf type theorems for some classes of fourth order semilinear elliptic and second order quasi-linear parabolic equations are proved.