Parabolic Inequalities Research Articles

Quasistatic Evolution of Damage in Thermo-electro-elastic-viscoplastic Materials With Normal Compliance

We have studied contact problem in a quasistatic process with boundary conditions, for which we couple a thermal effect and damage and electrical effect. We assume that the contact is with normal compliance in a form with a gap function and is associated to a slip rate-dependent version of Coulomb’s law of dry friction. We give the variational formulation, then the existence and the uniqueness of the weak solution. The proof is based on arguments of time-dependent variational inequalities, parabolic inequalities, and fixed points.

A Time-Fractional Parabolic Inequality on a Bounded Interval

We study a time-fractional parabolic inequality posed on a bounded interval and involving a wight function W, where the fractional derivative is considered in the Caputo sense. We establish a general condition ensuring that the set of weak solutions is empty. Next, some particular cases of the weight function W are discussed.

Variational and numerical analysis of a dynamic elasto-viscoplastic piezoelectric frictional contact problem

Abstract This work studies a mathematical model involving a dynamic contact between two elasto-viscoplastic piezoelectric bodies with damage. The contact is modeled on a combination of a normal compliance and a normal damped response law associated with friction. We derive a variational formulation of the problem and we prove the existence and uniqueness result of the weak solution. The proof is based on the classical existence and uniqueness result for parabolic inequalities, differential equations and fixed-point arguments. This paper examines a friction problem between two piezoelectric bodies that are elastic- viscoplastic. The study uses both numerical and analytical methods. The contact surface is assumed to have a thin layer of lubricant, and a damped response contact condition is considered. The problem also takes into account electrical and frictional effects. The paper proves the existence of a unique weak solution to the problem by using variational inequalities and a fixed point argument. Additionally, a fully discrete approximation is investigated using the Euler scheme and finite element method for the spatial variable and time derivatives. Error estimates are then calculated, leading to convergence results under certain regularity conditions.

Fujita type results for a quasilinear parabolic inequality of Hardy-Hénon type with time forcing terms

We investigate a quasilinear parabolic inequality of Hardy-Hénon type with a time forcing term ut−divA(x,u,Du)≥tγ|x|σuq,x∈RN,t>0,u(x,t)≥0,x∈RN,t>0,u(x,0)=u0(x)≥0,x∈RN,where Du represents the gradient of u, |x| represents the length of x∈RN; the function A is a Carathéodory function such that A(x,z,0)=0, A(x,z,w)⋅w≥0 for every (x,z,w)∈RN×[0,∞)×RN, and on the operator A we require weak m-coercivity, namely A(x,z,w)⋅w≥c0|x|β|A(x,z,w)|mm−1,(x,z,w)∈RN×[0,∞)×RNholds for some c0>0, m≥2 , and 0≤β<Nm−1; γ>−1, σ>−N, and q>N(m−1)N−β(m−1). Based on nonlinear capacity estimates, we obtain the Fujita type results to the above problem, i.e., the conditions on the parameters γ,σ,q such that the above problem admits no nontrivial nonnegative solutions for arbitrary initial value u0.

On the Absence of Weak Solutions of Nonlinear Nonnegative Higher Order Parabolic Inequalities with a Nonlocal Source

The paper proves the absence of solutions of semilinear parabolic inequalities and higher order systems with a singular potential and nonlocal sources. The proofs are based on the test function method developed by E. Mitidieri and S.I. Pokhozhaev.

On the Absence of Weak Solutions of Nonlinear Nonnegative Higher Order Parabolic Inequalities with a Nonlocal Source

On the Absence of Weak Solutions of Nonlinear Nonnegative Higher Order Parabolic Inequalities with a Nonlocal Source

Monotonicity and uniqueness of positive solutions to elliptic fractional p-equations

In this paper, we first obtain a maximum principle for parabolic fractional p-Laplacians. Then combining the maximum principle for parabolic fractional p-Laplacians with the parabolic inequalities, we give a new proof for monotonicity and uniqueness of solutions to elliptic fractional p-equations with gradient terms by the sliding method. In order to overcome the difficulties caused by the gradient term, we introduce some new tricks when applying the sliding method.We believe that parabolic inequalities and the maximum principle introduced here can be applied to many nonlinear nonlocal elliptic equations with gradient terms.

CONTACT PROBLEM WITH ADHESION AND WEAR IN ELECTRO-VISCOELASTICITY WITH DAMAGE

We consider a mathematical model which describes the dynamic process of con- tact between a piezoelectric body and two obstacles. The behavior of the material is modeled with a nonlinear electro-viscoelastic constitutive with law long memory and damage. The mechanical damage of the material, caused by excessive stess or strains, is described by a damage function whose evolution is modeled by an inclusion of parabolic type. The contact is modeled with adhesion and wear. The adhesion field, whose evolution is described by a first order differential equation. The evolution of the wear function is described with Ar- chard’s law. For the variational formulation of the contact problem, we present and prove the existence of a unique weak solution to the problem. The proof is based on arguments of time dependent variational inequalities, parabolic inequalities, differential equations and fixed point arguments.

Study of weak solutions for degenerate parabolic inequalities with nonstandard conditions

In this paper, we study the degenerate parabolic variational inequalities in a bounded domain. By solving a series of penalty problems, the existence and uniqueness of the solutions in the weak sense are proved by the energy method and a limit process.

Parabolic inequalities in inhomogeneous Orlicz-Sobolev spaces with gradients constraints and L 1-data

Abstract This work is devoted to the study of a new class of parabolic problems in inhomogeneous Orlicz spaces with gradient constraints and L 1-data. One proves the existence of the solution by studying the asymptotic behaviour as p goes to ∞, of a sequence of entropy solutions (up ) of some nonlinear parabolic equation in inhomogeneous Orlicz-Sobolev spaces with L 1-data involving the parameter p.

A Parabolic Harnack Principle for Balanced Difference Equations in Random Environments

We consider difference equations in balanced, i.i.d. environments which are not necessary elliptic. In this setting we prove a parabolic Harnack inequality (PHI) for non-negative solutions to the discrete heat equation satisfying a (rather mild) growth condition, and we identify the optimal Harnack constant for the PHI. We show by way of an example that a growth condition is necessary and that our growth condition is sharp. Along the way we also prove a parabolic oscillation inequality and a (weak) quantitative homogenization result, which we believe to be of independent interest.

Nonexistence criteria for systems of parabolic inequalities in an annulus

We are concerned with nonexistence results for a class of systems of parabolic inequalities in (0,∞)×A, where A={x∈RN:1<|x|≤2}. The considered systems involve a singular potential function V(x)=(|x|−1)−ρ, ρ>0, in front of the power nonlinearities. Two types of inhomogeneous boundary conditions on ∂B2={x∈RN:|x|=2} are discussed: Neumann type conditions and Dirichlet type conditions. Using a unified approach, an optimal criterium of nonexistence is obtained for both cases. Our study yields naturally optimal nonexistence results for the corresponding stationary systems.

VARIATIONAL ANALYSIS OF AN ELASTIC-THERMO-VISCOPLASTIC CONTACT PROBLEM WITH NORMAL DAMPED RESPONSE

We consider a quasistatic frictional contact problem between an elastic-viscoplastic body and an obstacle. The contact is modelled with normal damped response and a local friction law. The material is elastic-viscoplastic with two internal variables which may describe a temperature parameter and the damage of the contacting surface. We provide a variational formulation of the problem and prove the existence of a unique weak solution to the model. The proof is based on arguments of evolution equations with monotone operators, a classical existence and uniqueness result on parabolic inequalities and fixed point.

A Problem with Wear Involving Thermo-Electro-viscoelastic Materials

In this paper, we consider a mathematical model of a contact problem in thermo-electro- viscoelasticity. The body is in contact with an obstacle. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. We establish a variational formulation for the model and we prove the existence of a unique weak solution to the problem. The proof is based on a classical existence and uniqueness result on parabolic inequalities, differential equations and fixed point arguments. We present a variational formulation of the problem, and we prove the existence and uniqueness of the weak solution

A dynamic problem with wear involving electro-elastic-viscoplastic materials with damage

"A dynamic contact problem is considered in the paper. The material behavior is described by electro-elastic-viscoplastic law with piezoelectric effects. The body is in contact with damage and an obstacle. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. The damage of the material caused by elastic deformations. The evolution of the damage is described by an inclusion of parabolic type. The problem is formulated as a coupled system of an elliptic variational inequality for the displacement, variational equation for the electric potential and a parabolic variational inequality for the damage. We establish a variational formulation for the model and we prove the existence of a unique weak solution to the problem. The proof is based on a classical existence and uniqueness result on parabolic inequalities, differential equations and fixed point arguments."

Fujita type results for quasilinear parabolic inequalities with nonlocal terms

&lt;p style='text-indent:20px;'&gt;In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{cases} &amp;amp;u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\ &amp;amp;u(x,0) = u_0(x)\ge0 \,\, \text{ in } \mathbb R^N,\end{cases} \qquad (P^{\pm}) $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;where &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ u_0\in L^1_{loc}({\mathbb R}^N) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ L_{\mathcal{A}} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; denotes a weakly &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ m $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-coercive operator, which includes as prototype the &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ m $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Laplacian or the generalized mean curvature operator, &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ p,\,q&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, while &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ K\ast u^p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; stands for the standard convolution operator between a weight &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ K&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; satisfying suitable conditions at infinity and &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ u^p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. For problem &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ (P^-) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; we obtain a Fujita type exponent while for &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ (P^+) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; we show that no such critical exponent exists. Our approach relies on nonlinear capacity estimates adapted to the nonlocal setting of our problems. No comparison results or maximum principles are required.&lt;/p&gt;

Parabolic inequalities in Orlicz spaces with data in L 1

Abstract In this paper, we provide existence and uniqueness of entropy solutions to a general nonlinear parabolic problem on a general convex set with merely integrable data and in the setting of Orlicz spaces.

Existence of nonnegative solutions of nonlinear fractional parabolic inequalities

We study the existence of nontrivial nonlocal nonnegative solutions u(x, t) of the nonlinear initial value problems $$\begin{aligned} \left\{ \begin{array}{ll} (\partial _t -\Delta )^\alpha u\ge u^\lambda &{}\quad \mathrm{in}\,\, {\mathbb {R}}^n \times {\mathbb {R}},\,n\ge 1 \\ u=0 &{}\quad \mathrm{in}\,\, {\mathbb {R}}^n \times (-\infty ,0) \end{array}\right. \end{aligned}$$and $$\begin{aligned} \left\{ \begin{array}{ll} C_1 u^\lambda \le (\partial _t -\Delta )^\alpha u\le C_2 u^\lambda &{}\quad \mathrm{in}\,{\mathbb {R}}^n \times {\mathbb {R}},\,n\ge 1 \\ u=0 &{}\quad \mathrm{in}\, {\mathbb {R}}^n \times (-\infty ,0), \end{array}\right. \end{aligned}$$where \(\lambda ,\alpha ,C_1\), and \(C_2\) are positive constants with \(C_1 <C_2\). We use the definition of the fractional heat operator \((\partial _t -\Delta )^\alpha \) given in Taliaferro (J Math Pures Appl 133:287–328, 2020) and compare our results in the classical case \(\alpha =1\) to known results.

On Qualitative Properties of Sign-Constant Solutions of Some Quasilinear Parabolic Problems

We study the Cauchy problem for quasilinear parabolic inequalities containing squares of the first derivatives of an unknown function (the so-called nonlinearities of the KPZ type). The coefficients of the leading nonlinear terms of the inequalities considered either can be continuous functions (the regular case) or can admit power singularities (the singular case) of degree no greater than 1. For the regular case, we prove the damping of global nonnegative solutions to the problem studied. By damping, we mean the boundedness of the support of a solution for each positive t, uniform (with respect to t) convergence to zero as |x| → ∞, and vanishing (for any x) starting with a certain sufficiently large t. For the singular case, we proved that the problem considered has no global positive solutions.

Pricing of American Carbon Emission Derivatives and Numerical Method under the Mixed Fractional Brownian Motion

This paper studies the pricing of American carbon emission derivatives and its numerical method under the mixed fractional Brownian motion. To capture the long memory properties such as self-similarity and long-range dependence in the price process, we proposed a model based on a fractional Black–Scholes, which is more in line with the actual characteristics of the option market. We have outlined a power penalty approach using parabolic variation inequality and linear complementarity (LCP) which arises from mixed fractional Brownian motion. In addition, we introduced a nonuniform grid-based modification of the fitted finite volume method for numerical solution. Numerically, we show the impact of Hurst exponent on the pricing and analyze the convergence rates of the proposed penalty method. In conclusion, since mfBm is a well-developed mathematical model of strongly correlated stochastic processes, this model will be an efficient model for pricing carbon financial derivative.