Mixed problems for degenerate abstract parabolic equations and applications

In this paper, our primary focus is on studying the properties of self-similar solutions to the Cauchy problem. We specifically examine the behavior of these solutions in a double nonlinear time-dependent parabolic equation and their absorption in a non-homogeneous medium. Through the research of the topic matter, our aim is to deliver a more thorough comprehension of the finite speed perturbations propagation in the solution of the Cauchy problem for a nonlinear parabolic equation. This establishment of a property is essential to understand the dynamic nature of these equations. Furthermore, we delve into the self-similar analysis of the solution, which allows us to ascertain the condition of Fujita type global solvability for the Cauchy problem in a double nonlinear degenerate-type parabolic equation within a non-homogeneous medium. This analysis provides valuable insights into the behavior and potential solvability of these equations on a global mean. Additionally, we establish estimates for weak solutions depending on the growing density and the value of numerical parameters. By establishing these estimates, we provide a more comprehensive understanding of the behavior of the solutions in different scenarios.

Nonlinear oblique derivative problems for singular degenerate parabolic equations on a general domain

We consider a free boundary problem of a quasi-linear strongly degenerate parabolic equation arising from a model of pressure filtration of flocculated suspensions. We provide definitions of generalized solutions of the free boundary problem in the framework of L2 divergence-measure fields. The formulation of boundary conditions is based on a Gauss--Green theorem for divergence-measure fields on bounded domains with Lipschitz deformable boundaries and avoids referring to traces of the solution. This allows one to consider generalized solutions from a larger class than BV. Thus it is not necessary to derive the usual uniform estimates of spatial and time derivatives of the solutions of the correspondingregularized problem, as required by the BV approach. We first prove the existence and uniqueness of the solution of the regularized parabolic free boundary problem and then apply the vanishing viscosity method to prove the existence of a generalized solution to the degenerate free boundary problem.

Blow-up problems for nonlinear parabolic equations on locally finite graphs

<abstract><p>In this paper, an inverse problem of determining the space-dependent volatility from the observed market prices of options with different strikes is studied. Being different from other inverse volatility problem with classical parabolic equations, we apply the linearization method and introduce some variable substitutions to convert the original problem into an inverse source problem in a degenerate parabolic equation in a bounded area, from which an unknown volatility can be recovered and deficiencies caused by artificial truncation can be solved. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions are deduced, the uniqueness and stability of the minimizer are proved. Then, the Landweber iterative method is used to obtain a stable numerical solution of the inverse problem and some numerical experiments are also performed. The numerical results show that the algorithm which we proposed is robust and the unknown coefficient is recovered quite well.</p></abstract>

We study the extinction property of solutions to the Cauchy–Dirichlet problem for nonlinear parabolic equations of the order 2m with absorption potential in a semibounded cylinder (0,+∞) × Ω, where Ω is a bounded domain in ℝN, N ≥ 1. The sufficient conditions ensuring the extinction of a solution in a finite time, which depend on N,m, and q (where q is a parameter of the homogeneous nonlinearity in the main part of the equation), are obtained.

On the refined maximum principle for degenerate elliptic and parabolic problems

Optimal second-order regularity in the space variables is established for solutions to Cauchy–Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.

Linear and nonlinear degenerate differential operators and applications

We study boundary-value problems with Dirichlet conditions for nonlinear parabolic equations with variable delay (i.e., delay is a function of time) and degeneration at the initial time. The existence and uniqueness of the classical solution of this problem are proved. The a priori estimates of this solution are obtained.

For quasilinear doubly degenerate parabolic equations it has been possible to establish the existence of nonnegative generalized solutions to the first boundary-value problem that are Holder continuous up to the boundary of the cylinder QT=Ωx(0, T]. A typical example of an admissible equation is the equation of Newtonian polytropic filtration $$\frac{{\partial u}}{{\partial t}} - \frac{\partial }{{\partial x_i }}\left\{ {a_0 \left| u \right|^l \left| {u_x } \right|^{m - 2} u_{x_i } } \right\} = 0, a_0 > 0,l > 0,m > 2.$$ .

For a nonlinear degenerate parabolic equation, how to impose a suitable boundary value condition to ensure the well-posedness of weak solutions is a very important problem. It is well known that the classical Fichera-Oleinik theory has perfectly solved the problem for the linear case, and the optimal boundary value condition matching up with a linear degenerate parabolic equation can be depicted out by Fechira function. In this paper, a new method, which is called the weak characteristic function method, is introduced. By this new method, the partial boundary condition matching up with a nonlinear degenerate parabolic equation can be depicted out by an inequality from the diffusion function, the convection function, and the geometry of the boundary ∂Ω itself. Though, by choosing different weak characteristic function, one may obtain the differential partial boundary value conditions, an optimal partial boundary value condition can be prophetic. Moreover, the new method works well in any kind of the degenerate parabolic equations.

We prove the existence of suitably defined weak Radon measure-valued solutions of the homogeneous Dirichlet initial-boundary value problem for a class of strongly degenerate quasilinear parabolic equations. We also prove that: $$(i)$$ the concentrated part of the solution with respect to the Newtonian capacity is constant; $$(ii)$$ the total variation of the singular part of the solution (with respect to the Lebesgue measure) is nonincreasing in time. Conditions under which Radon measure-valued solutions of problem $$(P)$$ are in fact function-valued (depending both on the initial data and on the strength of degeneracy) are also given.

Coefficient inverse problems are reformulated to a unified integral differential equation. The presented method of conversion of the considered inverse problems to a unified Volterra integral-differential equation gives an opportunity to distribute the acquired results also to analogous inverse problems for non-linear parabolic equations of different types.

The purpose of this paper is to propose semidiscrete methods for approximating the solution of second order parabolic initial boundary value problems based on the penalty method of Babuška. The methods given here consist in discretizing, with respect to time, the parabolic equation and applying the penalty method to the resulting elliptic boundary value problems. It is shown, for a particular method, that the error is asymptotically of second order when measured in the $L_2 $-norm and is of order ${3 / 2}$ with respect to the Sobolev $H^1 $-norm. In contrast to semidiscrete Galerkin methods it is not required that the admissable approximating functions satisfy boundary conditions.