Class Of Nonlinear Parabolic Equations Research Articles

Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces

Abstract In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general Liouville-type and other global constancy results. Several applications and consequences are presented and discussed.

Hamilton and Li–Yau type gradient estimates for a weighted nonlinear parabolic equation under a super Perelman–Ricci flow

In this paper we derive elliptic and parabolic type gradient estimates for positive smooth solutions to a class of nonlinear parabolic equations on smooth metric measure spaces where the metric and potential are time dependent and evolve under a super Perelman–Ricci flow. A number of implications, notably, a parabolic Harnack inequality, a class of Hamilton type dimension-free gradient estimates and two general Liouville type theorems along with their consequences are discussed. Some examples and special cases are presented to illustrate the results.

Nonexistence of Global Solutions for a Class of Nonlinear Parabolic Equations on Graphs

Nonexistence of Global Solutions for a Class of Nonlinear Parabolic Equations on Graphs

Upper semicontinuity of pullback D$\mathcal {D}$‐attractors for nonlinear parabolic equation with nonstandard growth condition

AbstractThis paper is devoted to the well‐posed problem and the existence of pullback ‐attractors for a class of nonlinear parabolic equation with nonstandard growth condition. First, by making use of Galerkin's method and monotone operator method, the existence of solutions is proved in Orlicz–Sobolev space with variable exponents depending on time and space, then the uniqueness and continuity of solutions are also obtained. Finally, by verifying the pullback ‐asymptotic compactness, the existence of pullback ‐attractors is proved. In particular, the upper semicontinuity of pullback ‐attractors of the corresponding equation with respect to the disturbance parameter λ is also proved.

Equivalence of "generalized" solutions for nonlinear parabolic equations with variable exponents and diffuse measure data

We prove the equivalence of suitably defined weak solutions of a nonhomogeneous initial-boundary value problem for a class of nonlinear parabolic equations. We also develop the notion of both "renormalized" and "entropy" solutions with respect to the "generalized" $p(\cdot)$-capacity, initial datum, and diffuse measure data (which does not charge the set of null $p(\cdot)$-capacity). Conditions, under which "generalized weak" solutions of the nonhomogeneous problem are in fact well-defined, are also given.

Calderón-Zygmund estimates for nonlinear parabolic equations with matrix weights on nonsmooth domains

A wider class of nonlinear parabolic equations with degenerate/singular matrix weights is studied for Calderón-Zygmund type estimates of weak solutions in the setting of weighted Sobolev spaces. Minimal regularity assumptions on the associated nonlinearities as well as weights are investigated for such classical estimates.

Recovery of Nonlinear Terms for Reaction Diffusion Equations from Boundary Measurements

In the present article we study the inverse problem of determining a general semilinear term for a class of nonlinear parabolic equations. We derive a new criterion for the unique and stable recovery of general semilinear terms for these type of equations from the knowledge of the parabolic Dirichlet-to-Neumann map associated with solutions of the equation having zero initial data.

Existence of solutions to nonlinear parabolic equations via majorant integral kernel

We establish the existence of solutions to the Cauchy problem for a large class of nonlinear parabolic equations including fractional semilinear parabolic equations, higher-order semilinear parabolic equations, and viscous Hamilton–Jacobi equations by using the majorant kernel introduced in Ishige et al. (2020).

CONTROL DESIGN FOR A CLASS OF GENERAL NONLINEAR REACTION DIFFUSION EQUATIONS

<abstract> We consider a class of nonlinear parabolic equation with general source function $f(u)$, conduction function $g(u)$ and conduction coefficient $\rho(|\nabla u|^2)$ in multi-dimensional space. We establish new control conditions to guarantee that the positive solution exists globally. At the same time, under suitable control conditions, by means of the Sobolev inequality in multi-dimensional space, we obtain upper and lower bounds of the blow-up time $t^*$ in $\mathbb{R}^{n}\, (n\geqslant 2)$. Our work generalize the models, improve the method and remove the constraint of spatial dimension in many literatures. </abstract>

Anisotropic parabolic problem with variable exponent and regular data

In this paper, we study the existence of weak solutions for a class of nonlinear parabolic equations with regular data in the setting of variable exponent Sobolev spaces. We prove a "version" of a weak Lebesgue space estimate that goes back to "Lions J. L. Quelques méthodes de résolution des problèmes aux limites. Dunod, Paris (1969)" for parabolic equations with anisotropic constant exponents (pi(⋅)=pi).

Analysis of a quasi-reversibility method for nonlinear parabolic equations with uncertainty data

In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed—i.e., the solution does not depend continuously on the data. To regularize the instable solution, we develop some new methods to construct some new regularized solution. We also investigate the convergence rate between the regularized solution and the solution of our equations. In particular, we establish results for several equations with constant coefficients and time dependent coefficients. The equations with constant coefficients include the heat equation, extended Fisher–Kolmogorov equation, Swift–Hohenberg equation, and many others. The equations with time dependent coefficients include Fisher-type logistic equations, the Huxley equation, and the Fitzhugh–Nagumo equation. The methods developed in this paper can also be applied to get approximate solutions to several other equations including the 1-D Kuramoto–Sivashinsky equation, 1-D modified Swift–Hohenberg equation, strongly damped wave equation, and 1-D Burgers equation with randomly perturbed operator. Some numerical examples are given which illustrate the effectiveness of our method.

Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations

In this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows: {ut=αuxx+β[φ(u)]xx+f(u)inQ:=Ω×(0,T),u=0on∂Ω×(0,T),u(x,0)=u0(x)inΩ,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u_{t}=\\alpha u_{xx}+\\beta [\\varphi (u) ]_{xx}+f(u) &\\text{in} \\ Q:=\\Omega \\times (0,T), \\\\ u=0 &\\text{on} \\ \\partial \\Omega \\times (0,T), \\\\ u(x,0)=u_{0}(x) &\\text{in} \\ \\Omega , \\end{cases} $$\\end{document} where T>0, Omega subset mathbb{R} is a bounded interval, u_{0} is nonnegative bounded Radon measure on Ω, and alpha , beta geq 0, under suitable assumptions on φ and f. In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above. In particular, we study the asymptotic behavior of these Radon measure-valued solutions.

On a class of nonlinear parabolic equations with natural growth in non-reflexive Musielak spaces

An existence result of renormalized solutions for nonlinear parabolic Cauchy-Dirichlet problems whose model $$left{begin{array}{ll} displaystylefrac{partial b(x,u)}{partial t} -mbox{div}>mathcal{A}(x,t,u,nabla u)-mbox{div}> Phi(x,t,u)= f &mbox{ in }Omegatimes (0,T) b(x,u)(t=0)=b(x,u_0) & mbox{ in } Omega u=0 &mbox{ on } partialOmegatimes (0,T). end{array}right. $$ is given in the non reflexive Musielak spaces, where $b(x,cdot)$ is a strictly increasing $C^1$-function for every $xinOmega$ with $b(x,0)=0$, the lower order term $Phi$ is a non coercive Carath'{e}odory function satisfying only a natural growth condition described by the appropriate Musielak function $varphi$ and $f$ is an integrable data.

Stability and existence results for a class of nonlinear parabolic equations with three lower order terms and measure data using Lorentz spaces

We study both existence and stability of renormalized solutions for nonlinear parabolic problems with three lower order terms that have, respectively, growth with respect to u and to the gradient, whose model $$\begin{aligned}({\mathcal {P}})\ {\left\{ \begin{array}{ll} u_{t}-\varDelta _{p}u-\text {div}[c(t,x)|u|^{\gamma -1}u]+b(t,x)|\nabla u|^{\lambda }+d(t,x)|u|^{\iota }=\mu -\text {div}(E)\quad \text { in }Q,\\ u(0,x)=u_{0}(x) \quad \text { in }\varOmega ,\ u(t,x)=0\text { on }(0,T)\times \partial \varOmega , \end{array}\right. }\end{aligned}$$ where $$Q:=(0,T)\times \varOmega $$ (with $$\varOmega $$ is an open bounded subset of $${\mathbb {R}}^{N}$$ ( $$N\ge 2$$ ) and $$T>0$$ ), $$1<p<N$$ , $$\varDelta _{p}$$ is the usual p-Laplace operator, and $$\mu \in {\mathbf {M}}(Q)$$ is a (general) measure with bounded total variation on Q. As a consequence of our main results, we prove that the conditions $$\gamma =\frac{(N+2)(p-1)}{N+p}$$ , $$\lambda =\frac{N(p-1)+p}{N+2}$$ , $$0\le \iota \le p-\frac{N-p}{N}$$ , $$c\in L^{\tau =\frac{N+p}{p-1}}(Q)^{N}$$ , $$b\in L^{N+2,1}(Q)$$ and $$d\in L^{z',1}(Q)$$ (with $$z=\frac{pN-N-p}{\iota N}$$ ) are necessary and sufficient for the existence and the stability of solutions for every sufficiently regular $$u_{0}\in L^{2}(\varOmega )$$ , $$E\in L^{p'}(Q)^{N}$$ and irregular $$\mu \in {\mathbf {M}}(Q)$$ .

Singular Neumann Boundary Problems for a Class of Fully Nonlinear Parabolic Equations in One Dimension

In this paper, we discuss a singular Neumann boundary problem for a class of nonlinear parabolic equations in one space dimension. Our boundary problem describes the motion of a planar curve sliding along the boundary with a zero contact angle, which can be viewed as a limiting model for the capillary phenomenon. We study the uniqueness and existence of solutions by using the viscosity solution theory. We also show the convergence of the solution to a traveling wave as time proceeds to infinity when the initial value is assumed to be convex.

Stability properties of Radon measure-valued solutions for a class of nonlinear parabolic equations under Neumann boundary conditions

&lt;abstract&gt;&lt;p&gt;In this paper, we address the existence, uniqueness, decay estimates, and the large-time behavior of the Radon measure-valued solutions for a class of nonlinear strongly degenerate parabolic equations involving a source term under Neumann boundary conditions with bounded Radon measure as initial data.&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{equation*} \begin{cases} u_{t} = \Delta\psi(u)+h(t)f(x, t) \ \ &amp;amp;\text{in} \ \ \Omega\times(0, T), \\ \frac{\partial\psi(u)}{\partial\eta} = g(u) \ \ &amp;amp;\text{on} \ \ \partial\Omega\times(0, T), \\ u(x, 0) = u_{0}(x) \ \ &amp;amp;\text{in} \ \ \Omega, \end{cases} \end{equation*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ T &amp;gt; 0 $, $ \Omega\subset \mathbb{R}^{N}(N\geq2) $ is an open bounded domain with smooth boundary $ \partial\Omega $, $ \eta $ is an outward normal vector on $ \partial\Omega $. The initial value data $ u_{0} $ is a nonnegative bounded Radon measure on $ \Omega $, the function $ f $ is a solution of the linear inhomogeneous heat equation under Neumann boundary conditions with measure data, and the functions $ \psi $, $ g $ and $ h $ satisfy the suitable assumptions.&lt;/p&gt;&lt;/abstract&gt;

Existence of a renormalized solution of nonlinear parabolic equations with lower order term and general measure data

We give an existence result of a renormalized solution for a classof nonlinear parabolic equations@b(u)/@t div(a(x; t;grad(u))+ H(x; t;ru) = ,where the right side is a general measure, b is a strictly increasing C1-function,div(a(x; t;grad(u)) is a Leray{Lions type operator with growth in grad(u)and H(x; t;grad(u) is a nonlinear lower order term which satisfy the growth condition with respect to grad(u).

A class of nonlinear parabolic equations with anisotropic nonstandard growth conditions

In this paper, a doubly nonlinear parabolic problem involving nonstandard growth conditions under homogeneous Dirichlet boundary conditions is studied. To be a little precise, first, some energy estimates and inequalities are exploited to obtain energy decay estimates of the solution. Then, it is proved that the rest field u(x, t) = 0 is asymptotically stable in terms of natural energy associated with the solution. Second, we give two new blow-up criteria that one of them is obtained to remove the exponent restriction by using an extended form of the concavity method and the other one is a blow-up result of weak solutions with arbitrarily high initial energy. Moreover, an extinction result is also showed by establishing a new auxiliary lemma. The present results of this paper complement and improve a recent paper investigated by Liu et al. [J. Math. Phys. 59, 121504 (2018)].

A B-spline finite element method for solving a class of nonlinear parabolic equations modeling epitaxial thin-film growth with variable coefficient

In this paper, we propose an efficient B-spline finite element method for a class of fourth order nonlinear differential equations with variable coefficient. For the temporal discretization, we choose the Crank–Nicolson scheme. Boundedness and error estimates are rigorously derived for both semi-discrete and fully discrete schemes. A numerical experiment confirms our theoretical analysis.

Two-grid Raviart-Thomas mixed finite element methods combined with Crank-Nicolson scheme for a class of nonlinear parabolic equations

In this paper, we discuss a priori error estimates of two-grid mixed finite element methods for a class of nonlinear parabolic equations. The lowest order Raviart-Thomas mixed finite element and Crank-Nicolson scheme are used for the spatial and temporal discretization. First, we derive the optimal a priori error estimates for all variables. Second, we present a two-grid scheme and analyze its convergence. It is shown that if the two mesh sizes satisfy h = H2, then the two-grid method achieves the same convergence property as the Raviart-Thomas mixed finite element method. Finally, we give a numerical example to verify the theoretical results.