Pseudo-parabolic Problem Research Articles

ON THE EQUATION OF BARENBLATT–SOBOLEV

In this paper, we are interested in the following pseudoparabolic problem, known as the Barenblatt–Sobolev problem: f(∂ut) - Δu - ϵΔ∂ut = g with u(0, ⋅) = u0 where f is a non-monotone Lipschitz-continuous function, ϵ > 0 and [Formula: see text]. We show the existence of a critical value ϵ0 >0 such that: if ϵ > ϵ0, then the problem admits a unique solution; if ϵ = ϵ0, the solution is unique and it exists under an additional assumption on f; if ϵ < ϵ0, then the solution is not unique in general. Passing to the limit with ϵ to 0+, we prove the existence (and uniqueness) of the solution of the Barenblatt differential inclusion Δu + g ∈ f(∂ut) for a class of maximal monotone operators f. Next, we give an extension of the main result for a stochastic perturbation of the problem and we give some numerical illustrations of the Barenblatt and the Barenblatt–Sobolev equation.

Existence results for nonlinear pseudoparabolic problems

In this paper, we are interested in nonlinear pseudoparabolic problems of type: f ( t , x , ∂ t u ) − Div [ a ( x , u , ∂ t u ) ∇ u ] − Div [ b ( x , u , ∂ t u ) ∇ ∂ u t ] = g . f is a nondecreasing continuous function with respect to its third argument, a is bounded and b is positive and bounded. The result of existence is proved thanks to a time discretization scheme. Then, we derive some applications to the equation of Barenblatt, a degenerate case and differential inclusions. Finally, some numerical illustrations are proposed.

Solving nonlinear singular pseudoparabolic equations with nonlocal mixed conditions in the reproducing kernel space

We study the singular pseudoparabolic problems with nonlocal mixed conditions in the reproducing kernel space, in which all functions satisfy the nonlocal mixed conditions. Based on the reproducing kernel theorem, a very simple method for solving the singular pseudoparabolic problems with nonlocal mixed conditions is proposed. The final numerical experiment illustrates the method is efficient.

On a pseudoparabolic problem with constraint

This work deals with the study of a nonlinear degenerate pseudoparabolic problem. Arising from the modelling of sedimentary basin formation, the equation degenerates in order to take implicitly into account a constraint on the time derivative of the unknown. An existence result of a solution with an adapted compactness result and qualitative properties of the solutions are proposed (localization, finite speed of propagation, etc.).

Initial-boundary value problems for a class of pseudoparabolic equations with integral boundary conditions

In this paper, we deal with a class of pseudoparabolic problems with integral boundary conditions. We will first establish an a priori estimate. Then, we prove the existence, uniqueness and continuous dependence of the solution upon the data. Finally, some extensions of the problem are given.

Point Control with State Constraints

Controllability results are presented for pseudoparabolic problems with spacial domains in $\mathbb{R}^p $ with $p = 1,2$ or 3, as well as compatibility conditions for state constraints of noncontrollable problems. A concrete problem is considered and a regularity result is given for the optimal control.

Feedback operator for a pseudoparabolic optimal control problem

The feedback operator of a linear pseudoparabolic problem with quadratic criterion is obtained by decoupling of the optimality condition. The feedback operator is shown to be related to the solution of a Riccati equation formulated in theB*-algebra of bounded linear operators onL2(Ω). This approach shows that the linear feedback operator may be considered as a bounded operator fromL2(Ω) intoH01(Ω). Finally, we give a theorem establishing the convergence behavior for the feedback operators for these problems as they formally approach an analogous problem of parabolic type.

A bang-bang result for point control

In this note we consider the control of a pseudo-parabolic or parabolic problem at a point a in subject to constraints. We obtain a bang-bang property for an optimal control that depends on whether the components of a are irrational. We also show that such a result does not hold for the parabolic case p = 2 or 3.

Point control of pseudoparabolic problems

A control problem governed by a pseudoparabolic equation on Q = Ω × (0, T) where Ω is an open bounded set in R 2 or R 3 with a smooth boundary is studied. Here the controls are of the form v(x,t) = v(t) δ(x − a), a ɛ Ω. It is observed that if v ɛ L 2(0, T), then the trace of the corresponding solution y(., T; v) belongs to L 2(Ω). Existence, uniqueness, and regularity results are given for the optimal control as well as continuity results with respect to perturbation of the point a.

Formulation of pseudoparabolic boundary control problems

Boundary control of a pseudoparabolic problem is studied. The trace of a solution y( T; ϑ) corresponding to a control ϑ is shown to belong to L 2(Ω) and continuity properties are established. Two concrete examples are considered and regularity properties of the optimal control are determined in each case.

Convergence properties of optimal controls of pseudo-parabolic problems

Approximation properties of pseudo-parabolic optimal controls to the parabolic optimal control are considered for two concrete problems.

Point control: approximations of parabolic problems and pseudo parabolic problems

We consider quadratic control problems in which the underlying equation is of parabolic or pseudo-parabolic type and the distributed control is a Dirac-like function. The following approximating properties are studied i) the equation is fixed and the controls converge to a delta function and ii) the type of control is fixed and the pseudo-parahollc equations formally ,approach the parabolic

Existence and uniqueness of solutions of quasilinear transmission problems of both elliptic and pseudoparabolic type simulating oxygen transport in capillary and tissue

AbstractQuasilinear transmission problems of both the elliptic and pseudoparabolic type with differential operators in divergence form simulating oxygen transport in capillary and tissue are investigated. Using recent methods of the theory of pseudomonotone operators and singular perturbations (elliptic regularization) the existence of weak solutions for both problems has been proven. Moreover, the weak solution of the elliptic problem proves to be a solution in the classical sense. Finally, at most one classical solution of a slightly modified pseudoparabolic problem exists.

Uniform Error Estimates for Difference Approximations to NonLinear Pseudo-Parabolic Partial Differential Equations

Two difference-approximation schemes to a nonlinear pseudo-parabolic equation are developed. Each of these schemes is stable and possesses a unique solution which can be obtained by an iterative procedure. By means of a discrete energy estimate, each approximation is shown to converge in the uniform norm to the exact solution with a determined rate. It has also been found that for a fixed lattice, the solution to each of two difference approximations to a certain parabolic problem can be obtained as the limit of a sequence of difference approximations to a pseudo-parabolic problem.

Stability and convergence of difference approximations to pseudo-parabolic partial differential equations

Two difference approximations to the solution of a pseudo-parabolic problem are constructed and shown by means of stability analysis to converge in the "discrete" L 2 {L_2} norm. A relation between parabolic and pseudo-parabolic difference schemes is discussed, and the stability of difference approximations to backward time parabolic and pseudo-parabolic problems is also considered.