Parabolic Differential Inequalities Research Articles

On higher order elliptic and parabolic inequalities with singularities on the boundary

We prove the nonexistence of solutions to a number of higher order quasilinear elliptic and parabolic partial differential inequalities in bounded domains with point singularities on the boundary. The results are extended to systems of such inequalities. The proofs are based on the method of nonlinear capacity. We also present examples showing that the conditions obtained are sharp in the class of problems under consideration.

Elliptic and parabolic inequalities with point singularities on the boundary

It is shown that various quasilinear elliptic and parabolic differential inequalities and systems of such inequalities defined on bounded domains, and which have point singularities on the boundary do not have solutions. The method of nonlinear capacity is used in the proof. Examples show that the conditions obtained by this method cannot be improved in the class of problems under consideration.Bibliography: 14 titles.

Efficient numerical methods for pricing American options under stochastic volatility

AbstractFive numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two‐dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M‐matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear equations in the operator splitting method and in the penalty method are solved using a multigrid method. The projected multigrid method and the componentwise splitting method lead to a sequence of linear complementarity problems with one‐dimensional differential operators that are solved using the Brennan and Schwartz algorithm. The numerical experiments compare the accuracy and speed of the considered methods. The accuracies of all methods appear to be similar. Thus, the additional approximations made in the operator splitting method, in the penalty method, and in the componentwise splitting method do not increase the error essentially. The componentwise splitting method is the fastest one. All multigrid‐based methods have similar rapid grid independent convergence rates. They are about two or three times slower that the componentwise splitting method. On the coarsest grid the speed of the projected SOR is comparable with the multigrid methods while on finer grids it is several times slower. ©John Wiley & Sons, Inc. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Nonlinear parabolic differential equations and inequalities

The paper starts with a short review on the history of PDEs and their related inequalities. We present a unified approach to the basic parabolic differential inequalities. It starts in Section 2 with the famous Lemma of Nagumo and leads to recent and new result on parabolic equations with singular nonlinear elliptic operators (7) that include the p-Laplacian and the operator of capillary surfaces. Sections 3 and 4 deal with the one-dimensional case and radial solutions. A Semi-Strong Minimum Principle is found in Section 6.

Non-existence of solutions of semilinear parabolic differential inequalities in cones

Problems of the non-existence of global non-trivial solutions of semilinear parabolic differential inequalities and second-order systems in cones are investigated. The proofs are based on the method of test functions and do not use comparison theorems or the maximum principle.

A system of impulsive degenerate nonlinear parabolicfunctional‐differential inequalities

A theorem about a system of strong impulsive degenerate nonlinear parabolic functional‐differential inequalities in an arbitrary parabolic set is proved. As a consequence of the theorem, some theorems about impulsive degenerate nonlinear parabolic differential inequalities and the uniqueness of a classical solution of an impulsive degenerate nonlinear parabolic differential problem are established.

Impulsive nonlocal nonlinear parabolic differential problems

The aim of the paper is to prove a theorem about a weak impulsive nonlinear parabolic differential inequality together with weak impulsive nonlocal nonlinear inequalities. A weak maximum principle for an impulsive nonlinear parabolic differential inequality together with weak impulsive nonlocal nonlinear inequalities and an uniqueness criterion for the existence of the classical solution of an impulsive nonlocal nonlinear parabolic differential problem are obtained as a consequence of the theorem about the weak impulsive nonlinear parabolic differential inequality together with weak impulsive nonlocal nonlinear inequalities.

A regularity theory for a more general class of quasilinear parabolic partial differential equations and variational inequalities

A regularity theory for a more general class of quasilinear parabolic partial differential equations and variational inequalities

Exceptional Boundary Sets for Solutions of Parabolic Partial Differential Inequalities

Exceptional Boundary Sets for Solutions of Parabolic Partial Differential Inequalities

Exceptional boundary sets for solutions of parabolic partial differential inequalities

Let M \mathcal {M} be a second order, linear, parabolic partial differential operator with coefficients defined in a domain D = Ω × ( 0 , T ) \mathcal {D} = \Omega \times (0,\,T) in R n × R {{\mathbf {R}}^n} \times {\mathbf {R}} , with Ω \Omega a domain in R n {{\mathbf {R}}^n} . Let u u be a suitably regular real function in D \mathcal {D} such that u u is bounded below and M u \mathcal {M}u is bounded above in D \mathcal {D} . If u ⩾ 0 u \geqslant 0 on Ω × { 0 } \Omega \times \{ 0\} except on a set Γ × { 0 } \Gamma \times \{ 0\} , with Γ \Gamma a subset of Ω \Omega of suitably restricted Hausdorff dimension, then necessarily u ⩾ 0 u \geqslant 0 also on Γ × { 0 } \Gamma \times \{ 0\} . The allowable Hausdorff dimension of Γ \Gamma depends on the coefficients of M \mathcal {M} . For example, if M \mathcal {M} is the heat operator Δ − ∂ / ∂ t \Delta - \partial /\partial t , the Hausdorff dimension of Γ \Gamma needs to be smaller than the number of space dimensions n n . Analogous results are valid for exceptional boundary sets on the lateral boundary, ∂ Ω × ( 0 , T ) \partial \Omega \times (0,\,T) , of D \mathcal {D} .

The weak maximum principle for parabolic differential inequalities

Nous etablissons une form nouvelle du principe faible du maximum pour les solutions des inegalites differentielles du type parabolique.

Maximum principle for non-linear degenerate equations of the parabolic type

This paper establishes a weak maximum principle for the difference u − v of solutions to nonlinear degenerate parabolic differential inequality

On singular non-linear parabolic differential inequalities in unbounded domains

On singular non-linear parabolic differential inequalities in unbounded domains

Parabolic differential inequalities in cones

Theory of parabolic differential inequalities, flow-invariance of solutions and comparison theorems are discussed relative to a cone. In this paper we investigate the theory of parabolic differential inequalities in arbitrary cones. After discussing the fundamental results concerning parabolic inequalities in cones, we prove a result on flow-invariance which is then used to obtain a comparison theorem. This comparison result is useful in deriving upper and lower bounds on solutions of parabolic differential equations in terms of the solutions of ordinary differential equations. We treat the Dirichlet problem in this paper since its theory follows the general pattern of ordinary differential equations and requires less restrictive assumptions. The treatment of Neumann problem, on the other hand, demands stronger smoothness assumptions and depends heavily on strong maximum principle. The study of the corresponding results relative to Neumann problem is discussed elsewhere.

Unter- und Oberfunktionen und Differentialungleichungen zweiter Ordnung in normierten Räumen

An invariance theorem for a cone in a normed linear space with respect to solutions of general elliptic inequalities is obtained, which includes and generalizes monotonicity theorems for systems of second order ordinary, parabolic and elliptic differential inequalities.

Unboundedness of solutions of parabolic differential inequalities

Unboundedness of solutions of parabolic differential inequalities

On Existence and Nonexistence in the Large of Solutions of Parabolic Differential Equations with a Nonlinear Boundary Condition

This paper deals with solutions $u(t,x)$ of parabolic differential inequalities (a) $u_1 \leqq Lu$, or (b) $u_1 \geqq Lu$, respectively, where L is a linear, weakly elliptic differential operator of second order. The behavior of u for large t is studied under the assumption, that on the lateral boundary a nonlinear boundary condition of the form (a) ${{\partial u} / {\partial v}} \leqq f(u)$, or (b) ${{\partial u} / {\partial v}} \geqq f(u)$, is imposed, where $f(z) \to \infty $ as $z \to \infty $. It is shown that the value of the integral $\int^\infty {{dz} / {f(z)f'(z)}}$ is crucial for the growth properties of u. If this integral is infinite, then we have the case of global existence, i.e., any solution u of (a) is bounded in bounded sets. If, on the other hand, the integral is finite, then all solutions u of (b) with large initial values become infinite in finite time.

A Strong Maximum Principle for Quasilinear Parabolic Differential Inequalities

A maximum principle for ${C^1}$ solutions of quasilinear parabolic differential inequalities which retains the strong conclusion of Nirenberg’s well-known result [2] is established. The case of strongly differentiable solutions rather than of class ${C^1}$ is also discussed.

A strong maximum principle for quasilinear parabolic differential inequalities

A maximum principle for C 1 {C^1} solutions of quasilinear parabolic differential inequalities which retains the strong conclusion of Nirenberg’s well-known result [2] is established. The case of strongly differentiable solutions rather than of class C 1 {C^1} is also discussed.

On the maximum rate of decay of solutions of parabolic differential inequalities

On the maximum rate of decay of solutions of parabolic differential inequalities