Parabolic Differential Inequalities Research Articles

The Absence of Global Weak Solutions for a Quasilinear Parabolic Differential Inequality in Exterior Domain

The Absence of Global Weak Solutions for a Quasilinear Parabolic Differential Inequality in Exterior Domain

Fujita-type theorems for a nonlocal quasilinear parabolic differential inequality

Fujita-type theorems for a nonlocal quasilinear parabolic differential inequality

Strong unique continuation for variable coefficient parabolic operators with Hardy type potential

In this paper, we prove the strong unique continuation property at the origin for solutions of the following scaling critical parabolic differential inequality|div(A(x,t)∇u)−ut|≤M|x|2|u|, where the coefficient matrix A is Lipschitz continuous in x and t. Our main result sharpens a previous one of Vessella concerned with the subcritical case as well as extends an earlier result of one of us with Garofalo and Manna for the heat operator.

Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term

Abstract This work is concerned with the nonexistence of nontrivial nonnegative weak solutions for a quasilinear parabolic differential inequality with weighted nonlocal source term in the whole space, which involves weighted polytropic filtration operator or generalized mean curvature operator. We establish the new critical Fujita exponents containing the first and second types. The key ingredient of the technique in proof is the test function method developed by Mitidieri and Pohozaev. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.

Fujita Type Results for a Parabolic Differential Inequality with Weighted Nonlocal Source

Fujita Type Results for a Parabolic Differential Inequality with Weighted Nonlocal Source

On the critical behavior for inhomogeneous parabolic differential inequalities in the half-space

We consider the differential inequality with a nonlinear memory ∂tu−Δu≥1Γ(σ)∫0t(t−s)σ−1|u(s,x)|pds+w(x),t>0,x∈R+N, where N≥2, p>1, σ>0, and w⁄≡0. Under certain conditions on the initial value and the inhomogeneous term w, we show that the Fujita critical exponent is given by pσ∗=∞, for all σ>0. Next, we consider the limit case of the above problem as σ→0+, i.e. ∂tu−Δu≥|u|p+w(x),t>0,x∈R+N. A discontinuity phenomenon of the critical exponent is observed. Namely, we show that the Fujita critical exponent for the limit problem is equal to N+1N−1, which is different to limσ→0+pσ∗.

New Fujita type results for quasilinear parabolic differential inequalities with gradient dissipation terms

<abstract><p>This paper deals with the new Fujita type results for Cauchy problem of a quasilinear parabolic differential inequality with both a source term and a gradient dissipation term. Specially, nonnegative weights may be singular or degenerate. Under the assumption of slow decay on initial data, we prove the existence of second critical exponents $ \mu^{*} $, such that the nonexistence of solutions for the inequality occurs when $ \mu < \mu^{*} $.</p></abstract>

A unique continuation property for a class of parabolic differential inequalities in a bounded domain

This article is concerned with a strong unique continuation property of a forward differential inequality abstracted from parabolic equations proposed on a convex domain $ \Omega $ prescribed with some regularity and growth conditions. Our results show that the value of the solutions can be determined uniquely by its value on an arbitrary open subset $ \omega $ in $ \Omega $ at any given positive time $ T $. We also derive the quantitative nature of this unique continuation, that is, the estimate of a $ L^2(\Omega) $ norm of the initial data, which is majorized by that of solution on the bounded open subset $ \omega $ at terminal moment $ t = T $.

Blow-up results for a semilinear parabolic differential inequality in an exterior domain

Blow-up results for a semilinear parabolic differential inequality in an exterior domain

Nonexistence of solutions for the quasilinear parabolic differential inequalities with singular potential term and nonlocal source

This paper is devoted to proving some new nonexistence theorems for a class of quasilinear parabolic differential inequalities with a singular potential term and nonlocal source term in the case of homogeneous and non-homogeneous by the test function method.

Fujita type results for parabolic inequalities with gradient terms

In this paper we give some Fujita type results for strongly p-coercive quasilinear parabolic differential inequalities with both a diffusion term and a dissipative term, whose prototype is given by ut−Δpu≥a(x)uq−b(x)um|∇u|s in RN×R+, u≥0, u(x,0)=u0(x)≥0 in RN, where p>1, q>0, 0≤m<q, 0<s≤p(q−m)/(q+1) and a,b nonnegative weights which could be singular or degenerate. We prove the existence of Fujita type exponents qF, such that nonexistence of solutions for the inequality occurs when q<qF.

Nonexistence results for systems of parabolic differential inequalities in 2D exterior domains

Nonexistence results for systems of parabolic differential inequalities in 2D exterior domains

Nonexistence results for systems of elliptic and parabolic differential inequalities in exterior domains of ℝn

Nonexistence results for systems of elliptic and parabolic differential inequalities in exterior domains of ℝn

The absence of global positive solutions to semilinear parabolic differential inequalities in exterior domain

We are concerned with the absence of global positive solutions to certain semi-linear parabolic differential inequalities in exterior domains with nontrivial boundary condition. Assuming that the boundary value is positive, we determine the critical value of the exponent of a nonlinear term, which is responsible for the nonexistence of a solution. In the case of vanishing boundary value, the critical exponent was previously found by G. Laptev (2003) and has a smaller value.

Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds

We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the geometry of the underlying manifold, the power nonlinearity and the behavior of the potential at infinity.

A note on the classical weak and strong maximum principles for linear parabolic partial differential inequalities

In this note, we highlight a difference in the conditions of the classical weak maximum principle and the classical strong maximum principle for linear parabolic partial differential inequalities. We demonstrate, by the careful construction of a specific function, that the condition in the classical strong maximum principle on the coefficient of the zeroth-order term in the linear parabolic partial differential inequality cannot be relaxed to the corresponding condition in the classical weak maximum principle. In addition, we demonstrate that results (often referred to as boundary point lemmas) which conclude positivity of the outward directional derivatives of nontrivial solutions to linear parabolic partial differential inequalities at certain points on the boundary where a maxima is obtained cannot be obtained under the same zeroth-order coefficient conditions as in the classical strong maximum principle.

Extended weak maximum principles for parabolic partial differential inequalities on unbounded domains

In this paper, we establish extended maximum principles for solutions to linear parabolic partial differential inequalities on unbounded domains, where the solutions satisfy a variety of growth/decay conditions on the unbounded domain. We establish a conditional maximum principle, which states that a solution u to a linear parabolic partial differential inequality satisfies a maximum principle whenever a suitable weight function can be exhibited. Our extended maximum principles are then established by exhibiting suitable weight functions and applying the conditional maximum principle. In addition, we include several specific examples, to highlight the importance of certain generic conditions, which are required in the statements of maximum principles of this type. Furthermore, we demonstrate how to obtain associated comparison theorems from our extended maximum principles.

A Liouville comparison principle for solutions of semilinear parabolic inequalities in the whole space

Abstract. We obtain a new Liouville comparison principle for weak solutions (u,v) of semilinear parabolic second-order partial differential inequalities of the form u t - ℒ u - | u | q - 1 u ≥ v t - ℒ v - | v | q - 1 v ( * ) $u_t -{\mathcal {L}}u- |u|^{q-1}u\ge v_t -{\mathcal {L}}v- |v|^{q-1}v\qquad (*)$ in the whole space ℝ × ℝ n ${{\mathbb {R}} \times \mathbb {R}^n}$ . Here, n ≥ 1 ${n\ge 1}$ , q &gt; 1 ${q&amp;gt;1}$ and ℒ = ∑ i , j = 1 n ∂ ∂ x i a i j ( t , x ) ∂ ∂ x j , $ {\mathcal {L}}=\sum _{i,j=1}^n\frac{\partial }{{\partial }x_i}\biggl [ a_{ij}(t, x) \frac{\partial }{{\partial }x_j}\biggr ],$ where a i j ( t , x ) ${a_{ij}(t,x)}$ , i , j = 1 , ... , n ${i,j=1,\ldots ,n}$ , are functions that are defined measurable and locally bounded in ℝ × ℝ n ${{\mathbb {R}} \times \mathbb {R}^n}$ , and such that a i j ( t , x ) = a j i ( t , x ) ${a_{ij}(t,x)=a_{ji}(t,x)}$ and ∑ i , j = 1 n a i j ( t , x ) ξ i ξ j ≥ 0 $ \sum _{i,j=1}^n a_{ij}(t,x)\xi _i\xi _j\ge 0 $ for almost all ( t , x ) ∈ ℝ × ℝ n ${(t,x)\in {\mathbb {R}} \times \mathbb {R}^n}$ and all ξ ∈ ℝ n ${\xi \in \mathbb {R}^n}$ . We show that the critical exponents in the Liouville comparison principle obtained, which are responsible for the non-existence of non-trivial (i.e., such that u ¬ ≡ v ${{u\lnot \equiv v}}$ ) weak solutions to (*) in the whole space ℝ × ℝ n ${{\mathbb {R}} \times \mathbb {R}^n}$ , depend on the behavior of the coefficients of the operator ℒ ${\mathcal {L}}$ at infinity and coincide with those obtained for solutions of (*) in the half-space ℝ + × ℝ n ${{\mathbb {R}}_+\times {\mathbb {R}}^n}$ . As direct corollaries we obtain new Liouville-type theorems for non-negative weak solutions u of inequality (*) in the whole space ℝ × ℝ n ${{\mathbb {R}} \times \mathbb {R}^n}$ in the case when v ≡ 0 ${v\equiv 0}$ . All the results obtained are new and sharp.

Liouville theorems for a singular parabolic differential inequality with a gradient term

In this paper, we study proofs of some new Liouville theorems for a strongly p-coercive quasi-linear parabolic type differential inequality with a gradient term and singular variable coefficients. The proofs are based on the test function method developed by Mitidieri and Pohozaev.

On optimal arbitrage

In a Markovian model for a financial market, we characterize the best arbitrage with respect to the market portfolio that can be achieved using nonanticipative investment strategies, in terms of the smallest positive solution to a parabolic partial differential inequality; this is determined entirely on the basis of the covariance structure of the model. The solution is intimately related to properties of strict local martingales and is used to generate the investment strategy which realizes the best possible arbitrage. Some extensions to non-Markovian situations are also presented.