Non-uniformly Parabolic Equation Research Articles

General Boundary-Value Problem for Nonuniformly Parabolic Equations with Power Singularities

General Boundary-Value Problem for Nonuniformly Parabolic Equations with Power Singularities

Cauchy Problem for Nonuniformly Parabolic Equations with Power Singularities

Cauchy Problem for Nonuniformly Parabolic Equations with Power Singularities

Maximum principle for non-uniformly parabolic equations and applications

In this paper we study the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi's iteration. As applications, we show the existence of weak solutions for possibly degenerate stochastic differential equations with singular diffusion and drift coefficients. Moreover, by the Markov selection theorem of Krylov [8], we also establish the existence of the associated strong Markov family.

Global dynamics of a nonlocal non-uniformly parabolic equation arising from the curvature flow* *The first author is supported by NSF of China (No. 12271437 and No. 12071203). The second author is supported by NSF of China (No. 11971498) and NSF of Guangdong Province (No. 2019A1515011339). The third author is supported by NSF of China (No. 11871148).

This paper studies a type of non-uniformly parabolic problem with nonlocal term ut=up(uxx+u−u¯)0<t<Tmax,0<x<a,ux(t,0)=ux(t,a)=00<t<Tmax,u(0,x)=u0(x)0<x<a, where p > 1, a > 0. First the classification of the finite-time blow-up/global existence phenomena based on the associated energy functional and explicit expression of all nonnegative steady states are demonstrated. More importantly, we derive that any bounded solution converges to some steady state as t → +∞. The difficulties in proving this convergence result lie in the existence of a continuum of steady states and the lack of the comparison principle due to the introduction of nonlocal term. To conquer these difficulties, we combine the applications of Lojasiewicz–Simon inequality and energy estimates.

The Inverse Problem of Recovering the Source Function in a Multidimensional Nonuniformly Parabolic Equation

The Inverse Problem of Recovering the Source Function in a Multidimensional Nonuniformly Parabolic Equation

General boundary value problem for nonuniformly parabolic equations with power singularities

General boundary value problem for nonuniformly parabolic equations with power singularities

Non-uniformly parabolic equations and applications to the random conductance model

We study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on mathbb Z^d. In particular, we provide an oscillation decay assuming only certain summability properties of the conductances and their inverse, thus improving recent results in that direction. As an application, we provide a local limit theorem for the random walk in a random degenerate and unbounded environment.

Cauchy problem for non-uniformly parabolic equations with power singularities

The Cauchy problem for non-uniformly 2b-parabolic equations with degenerations is investigated. Coefficients of parabolic equations can have power singularities of arbitrary order with respect to any variables on some set of points. Using prior estimates and Arzelа’s and Riesz’s theorems the existence and integral representation of the unique solution to the formulated Cauchy problem are established. Estimates for the solution of the Cauchy problem and its derivatives in Hölder spaces with power weight are found. The order of the power weight is defined in terms of orders of the power singularities and degenerations in coefficients of 2b-parabolic equations. Cite as: І. P. Medynsky, “Fundamental solutіons for degenerate parabolіc equatіons: existence, propertіes and some theіr applіcatіons,” Mat. Met. Fiz.-Mekh. Polya , 64 , No. 2, 31–41 (2021) (in Ukrainian), https://doi.org/10.15407/mmpmf2021.64.2.31-41

General initial data for a class of parabolic equations including the curve shortening problem

<p style='text-indent:20px;'>The Cauchy problem for a class of non-uniformly parabolic equations including (4) is studied for initial data with less regularity. When <inline-formula><tex-math id="M1">\begin{document}$ m\in(1,2] $\end{document}</tex-math></inline-formula>, it is shown that there exists a smooth solution for <inline-formula><tex-math id="M2">\begin{document}$ t&gt;0 $\end{document}</tex-math></inline-formula> when the initial data belongs to <inline-formula><tex-math id="M3">\begin{document}$ L_{\text{loc}}^p, p&gt;1 $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M4">\begin{document}$ m&gt;2 $\end{document}</tex-math></inline-formula>, the same results holds when the initial data belongs to <inline-formula><tex-math id="M5">\begin{document}$ W_{\text{loc}}^{1,p}, p\geq m-1 $\end{document}</tex-math></inline-formula>. An example is displayed to show that a smooth solution may not exist when the initial data is merely in <inline-formula><tex-math id="M6">\begin{document}$ L^p_{\text{loc}}, p&gt;1 $\end{document}</tex-math></inline-formula>. Solvability of the weak solution is also studied.

Inverse source problem for parabolic equation with the condition of integral observation in time

Abstract We consider the inverse problem of source determination in nonuniformly parabolic equation under the additional condition of integral observation. We investigate the questions of existence and uniqueness of solution. Two types of sufficient conditions for the unique solvability of the inverse problem are obtained. Examples of inverse problems are given for which the conditions of the proved theorems are fulfilled.

A new proof of the existence of weak solutions to a model for phase evolution driven by material forces

We prove the existence of weak solutions to a one-dimensional initial-boundary value problem for a model system of partial differential equations, which consists of a sub-system of linear elasticity and a nonlinear non-uniformly parabolic equation of second order. To simplify the existence proof of weak solutions in the 2006 paper of Alber and Zhu, we replace the function |p|κ:=|p|2/|p|2+κ2 in that work by |p|κ:=|p|2+κ2. The model is formulated by using a sharp interface model for phase transformations that are driven by material forces. Copyright © 2017 John Wiley & Sons, Ltd.

The well-posedness of renormalized solutions for a non-uniformly parabolic equation

The well-posedness of renormalized solutions for a non-uniformly parabolic equation

Strong solutions and the initial data space for some non-uniformly parabolic equations

The solution for this problem can be considered in different meanings, for example, a classical solution, strong solution, weak (or mild) solution, entropy solution, etc. This paper is devoted to a strong solution for this problem. If A is an unbounded linear operator in Hilbert space H, then it is proved that there does not exist a strong solution for all u0 ∈ H (see [2, 11]). It is proved in [13] that there exists a strong solution if and only if u0 ∈ [D(A), H]1/2, where [D(A), H]1/2 is an interpolation space (see [9,15]). If A is a nonlinear operator, the weak solutions of these problems is often investigated. We consider the case where the operator A has the form:

Renormalized solutions for a non-uniformly parabolic equation

In this paper we prove the existence of nonnegative renormalized solutions for the initial-boundary value problem of a non-uniformly parabolic equation. Some well-known parabolic equations are the special cases of this equation.

Exact Traveling Wave Solutions to a Model for Solid-Solid Phase Transitions Driven by Configurational Forces

This paper studies the exact traveling wave solutions to a model for solid-solid phase transitions driven by configurational forces. The model consists of the partial differential equations of linear elasticity coupled to a quasilinear nonuniformly parabolic equation of second order, which describes the diffusionless phase transitions of solid materials. By using the hyperbolic tangent function expansion method and homogeneous balance method, some exact traveling wave solutions, including solitary wave solutions are obtained for the phase transitions model in one space dimension.

Spherically symmetric solutions to a model for phase transitions driven by configurational forces

We prove the global-in-time existence of spherically symmetric solutions to an initial-boundary value problem for a system of partial differential equations, which consists of the equations of linear elasticity and a nonlinear, non-uniformly parabolic equation of second order. This problem models the evolutional behavior of materials in which martensitic phase transitions, driven by configurational forces, take place. Moreover, it can be considered to be a regularization of the corresponding sharp interface model. By assuming that the solutions are spherically symmetric, we reduce the original multi-dimensional problem to the one in one space dimension, then prove the existence of spherically symmetric solutions. Our proof is valid due to the essential feature that the resulting problem is one-dimensional.

Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity

We consider the quasilinear parabolic–parabolic Keller–Segel system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a convex smooth bounded domain Ω ⊂ R n with n ⩾ 1 . It is proved that if S ( u ) D ( u ) ⩽ c u α with α < 2 n and some constant c > 0 for all u > 1 , then the classical solutions to the above system are uniformly-in-time bounded, provided that D ( u ) satisfies some technical conditions such as algebraic upper and lower growth (resp. decay) estimates as u → ∞ . This boundedness result is optimal according to a recent result by the second author (Winkler, 2010 [27]), which says that if S ( u ) D ( u ) ⩾ c u α for u > 1 with c > 0 and some α > 2 n , n ⩾ 2 , then for each mass M > 0 there exist blow-up solutions with mass ∫ Ω u 0 = M . In addition, this paper also proves a general boundedness result for quasilinear non-uniformly parabolic equations by modifying the iterative technique of Moser–Alikakos (Alikakos, 1979 [1]).

Entropy solutions for a non-uniformly parabolic equation

In this paper we prove the existence and uniqueness of entropy solutions for the initial-boundary value problem of a non-uniformly parabolic equation. Moreover, we establish a comparison result. Some well-known parabolic equations are the special cases of this equation.

Existence and uniqueness of weak solutions for a non-uniformly parabolic equation

In this paper we develop a unifying method to prove the existence and uniqueness of weak solutions for the initial–boundary value problem of a non-uniformly parabolic equation. Some well-known parabolic equations are the special cases of this equation.

On non-uniformly parabolic functional differential equations

We consider initial boundary value problems for second order quasilinear parabolic equations where also the main part contains functional dependence on the unknown function and the equations are not uniformly parabolic. The results are generalizations of that of [10]