In the case with zero initial energy Zhou [11] obtained a blow-up result for a nonlinear wave equation in.
Then Messaoudi [20] obtained the global existence of solutions for the viscoelastic equation, at same time he also obtained a blow-up result with negative energy.
We obtain a blow-up criterion of smooth solutions to (1.2), which improves our previous result (see [2]).
In order to obtain a blow-up condition corresponding to (1.1), we have to modify the function e − ε | x | 2 used in [31 33], and introduce a test function ϕ ε ( x ) as follows: ϕ ε ( x ) = A ε e − ε | x | with A ε = 1 ∫ Ω e − ε | x | d x.
In this paper, we will adapt the method of Chen et al.[13] to establish the local well-posedness for the incompressible porous media equation (1.1) and to obtain a blow-up criterion of smooth solutions in the framework of Triebel-Lizorkin spaces.
Under suitable assumptions on the initial data and the relaxation function g, we obtain a blow-up result for the solution with negative initial energy and some positive initial energy if (p>rho+2), and get a global existence result for any initial data if (pleqrho+2) using the perturbed energy functional technique.
Furthermore, some authors extended the above works for the semilinear case (1.2) to degenerate reaction-diffusion equations involving a nonlinear memory term and obtained a corresponding blow-up analysis (see, for example, [12 15]).
Kafini and Messaoudi [6] considered a nonlinear wave equation and obtained a finite-time blow-up result with arbitrary positive initial energy.
end{aligned} (1.3) They obtained a lower bound for the blow-up time (t^) if the blow-up does really occur together with a criterion for getting a blow-up.
Following the proof of Lemma 2.9 given by Zhao et al. in [21], we can obtain a similar blow-up result of the solution to the Cauchy problem (2.2).
Yuan [8] obtained a Beale-Kato-Majda type blow-up criterion for a smooth solution ( u, v, b ) to the Cauchy problem for (1.1) that relies on the vorticity of velocity ∇ × u only.
