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.
This paper deals with a Kirchhoff type equation with variable exponent nonlinearities, subject to a nonlinear boundary condition. Under appropriate conditions and regarding arbitrary positive initial energy, it is proved that solutions blow up in a finite time. Moreover, we obtain the upper bound estimate of the blow-up time.
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.
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