Global existence and blow up of solutions for pseudo-parabolic equation with singular potential

Abstract In this paper, the initial boundary value problem for a nonlocal semilinear pseudo-parabolic equation is investigated, which was introduced to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved. The existence, uniqueness and asymptotic behavior of the global solution and the blowup phenomena of solution with subcritical initial energy are established. Then these results are extended parallelly to the critical initial energy. Further the blowup phenomena of solution with supercritical initial energy is proved, but the existence, uniqueness and asymptotic behavior of the global solution with supercritical initial energy are still open.

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Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation

In this paper we consider the semilinear wave equation with logarithmic nonlinearity. By modifying and using potential well combined with logarithmic Sobolev inequality, we derive the global existence and infinite time blow up of the solution at low energy level E ( 0 ) d . Then these results are extended in parallel to the critical case E ( 0 ) = d . Besides, with additional assumptions on initial data, the infinite time blow up result is given with arbitrary positive initial energy E ( 0 ) > 0 .

Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity

In this paper we consider the semi-linear wave equation with logarithmic nonlinearity. By modifying and using potential well combined with Sobolev inequality, we gain the results of global existence and blow up at infinity of solution at all initial energy level. The results in this article guide that the polynomial nonlinearity is a critical condition of blow up in finite time for the semi-linear wave equation.

Global existence and blow up of solution for semi-linear hyperbolic equation with logarithmic nonlinearity

In this article, we investigate the initial-boundary value problem for a class of finitely degenerate semilinear parabolic equations with singular potential term. By applying the Galerkin method and Banach fixed theorem we establish the local existence and uniqueness of the weak solution. On the other hand, by constructing a family of potential wells, we prove the global existence, the decay estimate and the finite time blow-up of solutions with subcritical or critical initial energy.

Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials

The initial boundary value problem of a class of reaction-diffusion systems (coupled parabolic systems) with nonlinear coupled source terms is considered in order to classify the initial data for the global existence, finite time blowup and long time decay of the solution. The whole study is conducted by considering three cases according to initial energy: the low initial energy case, critical initial energy case and high initial energy case. For the low initial energy case and critical initial energy case the sufficient initial conditions of global existence, long time decay and finite time blowup are given to show a sharp-like condition. In addition, for the high initial energy case the possibility of both global existence and finite time blowup is proved first, and then some sufficient initial conditions of finite time blowup and global existence are obtained, respectively.

Global well-posedness of coupled parabolic systems

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Global well-posedness of coupled parabolic systems