Cauchy problems of semilinear pseudo-parabolic equations

Abstract

This paper concerns with the Cauchy problems of semilinear pseudo-parabolic equations. After establishing the necessary existence, uniqueness and comparison principle for mild solutions, which are also classical ones provided that the initial data are appropriately smooth, we investigate large time behavior of solutions. It is shown that there still exist the critical global existence exponent and the critical Fujita exponent for pseudo-parabolic equations and that these two critical exponents are consistent with the corresponding semilinear heat equations.

A PRIORI L2ERROR ANALYSIS FOR AN EXPANDED MIXED FINITE ELEMENT METHOD FOR QUASILINEAR PSEUDO-PARABOLIC EQUATIONS

“…In this paper we discussed the quasilinear pseudo-parabolic equations with a locally Lipschitz continuous f (u) which are important in many practical applications as shown in [5,9,11]. Applying an expanded mixed finite element method we constructed the approximations of the scalar unknown, its flux and its gradient, directly.…”

“…This type of equation which has the mixed derivative term with respect to temporal and spatial variables is called as a pseudo-parabolic equation. It represents physical phenomena arising in the various areas such as in the flow of fluids through fissured materials [3], thermodynamics [7], semiconductor [5] and other applications. For details about the physical significance and various properties of the existence and uniqueness of the solutions of the pseudo-parabolic equations we refer to [3,5,6,7,9,11,24].…”

Cauchy problems of pseudo-parabolic equations with inhomogeneous terms

“…Since the last century, pseudo-parabolic equations have been studied in different aspects, such as the integral representations of solutions [12], long-time behavior of solutions [17], Riemann problem and Riemann-Hilbert problem [8], nonlocal boundary value problems [4], and periodic problems [6,18,24,25]. Cao et al [5] investigated the blow-up theorems of the Cauchy problem…”

“…As mentioned in [5,39], in studying critical exponents for parabolic equations, the main method is to construct global self-similar supersolutions and blowing-up self-similar subsolutions, and most studies used this method. Nevertheless, owing to the appearance of the third-order term kΔu t , it is almost impossible to use the method constructing supersolutions and subsolutions for the pseudo-parabolic equations (1.2) or 1.3.…”

Extrapolated Expanded Mixed Finite Element Approximations of Semilinear Sobolev Equations

“…The problem (1.1)-(1.3) arises in the various areas, for examples, in the flow of fluids through fissured materials [1] and thermodynamics [4]. For details about the physical significance and the existence and uniqueness of the solutions of the Sobolev equations, we refer to [1,2,3,4,6,7,12].…”