Degenerate Parabolic Equations Research Articles

On Fundamental Solutions and Gaussian Bounds for Degenerate Parabolic Equations with Time-dependent Coefficients

On Fundamental Solutions and Gaussian Bounds for Degenerate Parabolic Equations with Time-dependent Coefficients

Total variation regularization analysis for inverse volatility option pricing problem

We investigate in this paper an inverse problem of recovering the space-dependent volatility in option pricing. To enhance precision across the domain, we transform the original problem into an inverse source problem of a bounded degenerate parabolic equation, utilizing linearization and variable substitution. Unlike classical methods, we apply a total variation regularization combined with a novel generalized finite integration technique. This approach accommodates volatility jumps in an overnight rate scenario. Leveraging an optimal control framework, we demonstrate that the inverse problem can be reformulated as an optimal control problem whose existence, necessary conditions, local uniqueness and stability for the minimizer of control functional are obtained. For numerical verification, we derive the Euler equation and design a discretization algorithm with generalized finite integration technique. Numerical examples showcase the robustness of our approach, highlighting advantages in accuracy and effectiveness over other strategies.

Bilinear optimal control of a one‐dimensional degenerate parabolic equation with a nonlocal term

In this paper, a one‐dimensional degenerate parabolic equation involving a multiplicative control with a nonlocal space term represented by an integral kernel is considered. After some well‐posedness results, we derive several properties of the control‐to‐state mapping. We establish the existence of at least one optimal control and derive the first and the second conditions for optimality, giving the local uniqueness of an optimal control. Finally, we prove that we may expect the global uniqueness of the optimal control under additional assumptions.

Local C0,1-Regularity for the Parabolic p-Laplacian Equation on the Group SU(3)

In this article, when 2≤p≤4, we establish the Cloc0,1-regularity of weak solutions to the degenerate parabolic p-Laplacian equation ∂tu=−∑i=16Xi*(|∇Hu|p−2Xiu) on the group SU(3) granted with horizontal vector fields X1, …, X6. Compared to the Heisenberg group, Hn, we obtained the optimal range of p; that is, 2≤p≤4.

Existence and regularity results for nonlinear anisotropic degenerate parabolic equations

Existence and regularity results for nonlinear anisotropic degenerate parabolic equations

Boundary controllability for a 1D degenerate parabolic equation with a Robin boundary condition

AbstractIn this paper, we prove the null controllability of a one-dimensional degenerate parabolic equation with a weighted Robin boundary condition at the left endpoint, where the potential has a singularity. We use some results from the singular Sturm–Liouville theory to show the well-posedness of our system. We obtain a spectral decomposition of a degenerate parabolic operator with Robin conditions at the endpoints, we use Fourier–Dini expansions and the moment method introduced by Fattorini and Russell to prove the null controllability and to obtain an upper estimate of the cost of controllability. We also get a lower estimate of the cost of controllability by using a representation theorem for analytic functions of exponential type.

On the weak Harnack inequality for unbounded non-negative super-solutions of degenerate double-phase parabolic equations

We give a proof of the weak Harnack inequality for non-negative super-solutions of degenerate double-phase parabolic equations under the additional assumption that u∈Llocs(ΩT) with some sufficiently large s.

Curve shortening flow on Riemann surfaces with conical singularities

AbstractWe study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic equation. In this case, we establish short time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and obtain some contracting and convergence results.

Asymptotic behaviour for a parabolic p-Laplacian equation with an advection force

The rescaling method is presented to allow us establishing the interface functions and the nonnegative local solutions to the evolution of nonlinear degenerate parabolic p-Laplacian process with conservation laws that are posed in one-dimensional space. This equation has a self-similar solution which represents the main feature of this work. The Cauchy problem (CP) for this equation with specific restrictions in the range of parameters and the negative advection coefficient is considered. In this work, there are several regions to discuss the qualitative analysis for the local weak solutions and the asymptotic interfaces in the irregular domains. The dominating of the advection force over the p-Laplacian type diffusion will appear clearly in these regions. Moreover, the solutions of the CP for the degenerate parabolic p-Laplacian advection equations are asymptotically equal to the solutions of advection equations under some restrictions.

Nonlinear Degenerate Parabolic Equations with a Singular Nonlinearity

Nonlinear Degenerate Parabolic Equations with a Singular Nonlinearity

Fujita-type results for the degenerate parabolic equations on the Heisenberg groups

Fujita-type results for the degenerate parabolic equations on the Heisenberg groups

Computational modeling of early-stage breast cancer progression using TPFA method: A numerical investigation

In this paper, a finite volume (TPFA) method is employed to simulate a degenerate breast cancer model that captures the progressive mutations from a normal breast stem cell to a tumor cell. The model incorporates a degenerate parabolic equation to represent the interaction between solid tumor growth and its environment, which involves the release of degradative enzymes governed by a partial differential equation. The discrete maximum principle is verified to be satisfied by the proposed finite volume scheme, and the convergence of the existing discrete solutions to a weak solution is proven. Additionally, the development of breast cancer is demonstrated through a numerical experiment.

Dynamical behavior of a degenerate parabolic equation with memory on the whole space

This paper is concerned with the existence and uniqueness of global attractors for a class of degenerate parabolic equations with memory on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{n}$\\end{document}. Since the corresponding equation includes the degenerate term div{a(x)∇u}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\operatorname{div}\\{a(x)\ abla u\\}$\\end{document}, it requires us to give appropriate assumptions about the weight function a(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$a(x)$\\end{document} for studying our problem. Based on this, we first obtain the existence of a bounded absorbing set, then verify the asymptotic compactness of a solution semigroup via the asymptotic contractive semigroup method. Finally, the existence and uniqueness of global attractors are proved. In particular, the nonlinearity f satisfies the polynomial growth of arbitrary order p−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p-1$\\end{document} (p≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p\\geq 2$\\end{document}) and the idea of uniform tail-estimates of solutions is employed to show the strong convergence of solutions.

On anisotropic parabolic equation with nonstandard growth order

ABSTRACT In this paper, the existence and the uniqueness of an evolutionary anisotropic p i ( x ) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic p ( x ) − Laplace equation with convection term via L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc.

Observability Inequality from Measurable Sets for Degenerate Parabolic Equations and its Applications

Observability Inequality from Measurable Sets for Degenerate Parabolic Equations and its Applications

High-order weighted compact nonlinear scheme for solving degenerate parabolic equations

High-order weighted compact nonlinear scheme for solving degenerate parabolic equations

Second-order optimality conditions for the bilinear optimal control of a degenerate equation

The main purpose of this paper is the study of second-order optimality conditions for the bilinear control of a strongly degenerate parabolic equation. The equation is degenerate at the boundary of the spatial domain. The well-posedness of the state equation, as well as weak maximum principles are established. We prove some differentiability properties of the control-to-state operator and the existence of optimal solutions. Finally, we derive first- and second-order optimality conditions for the system.

Classification of nonnegative traveling wave solutions for certain 1D degenerate parabolic equation and porous medium type equation

Classification of nonnegative traveling wave solutions for certain 1D degenerate parabolic equation and porous medium type equation

Interior Holder Regularity and Infinite Time Extinction to a doubly Nonlinear Degenerate Parabolic Equations of Signed Solutions

We study doubly nonlinear parabolic equation with sign changing solutions. We established the Holder regularity and the infinite time extinction of it within a parabolic domain. GANIT J. Bangladesh Math. Soc. 43.1 (2023) 75- 91

Justification of the exact asymptotics of the fundamental solution for a degenerate parabolic equation with a small parameter

AbstractIn this paper, we propose a scheme for constructing the asymptotics of the fundamental solution of a linear degenerate parabolic equation with a small parameter. The asymptotics is constructed using the operator representation of the Dirac delta function and the non‐oscillating Wentzel‐Kramers‐Brillouin (WKB) method. In addition, a method to justify the obtained asymptotics by proving the convergence of the formal series that arises when using the WKB method is shown. Several examples of degenerate parabolic equations are considered, for which a fundamental solution is constructed, and its main properties are indicated.