Class Of Parabolic Equations Research Articles

On a Fractional Parabolic Equation with Regularized Hyper-Bessel Operator and Exponential Nonlinearities

Recent decades have witnessed the emergence of interesting models of fractional partial differential equations. In the current work, a class of parabolic equations with regularized Hyper-Bessel derivative and the exponential source is investigated. More specifically, we examine the existence and uniqueness of mild solutions in Hilbert scale-spaces which are constructed by a uniformly elliptic symmetry operator on a smooth bounded domain. Our main argument is based on the Banach principle argument. In order to achieve the necessary and sufficient requirements of this argument, we have smoothly combined the application of the Fourier series supportively represented by Mittag-Leffler functions, with Hilbert spaces and Sobolev embeddings. Because of the presence of the fractional operator, we face many challenges in handling proper integrals which appear in the representation of mild solutions. Besides, the source term of an exponential type also causes trouble for us when deriving the desired results. Therefore, powerful embeddings are used to limit the growth of nonlinearity.

Solvability and dynamics study of a weak solution for a nonlinear parabolic problem with the bessel operator

In this paper, we present a class of parabolic equation with nonlinear where we show two part of this study the theoretical part we prove the existence and uniqueness of the solution by energy inequality method. Then we present finite time blow up by using the energy method.

Regularizing effect and decay results for a parabolic problem with repulsive superlinear first order terms

We want to analyze both regularizing effect and long, short time decay concerning a class of parabolic equations having first order superlinear terms. The model problem is the following: $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle u_t-\text {div }(A(t,x)|\nabla u|^{p-2}\nabla u)=\gamma |\nabla u|^q &{} \text {in}\,\,(0,T)\times \Omega ,\\ u=0 &{}\text {on}\,\,(0,T)\times \partial \Omega ,\\ u(0,x)=u_0(x) &{}\text {in}\,\, \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega $$ is an open bounded subset of $${{\,\mathrm{{{\mathbb {R}}}}\,}}^N$$ , $$N\ge 2$$ , $$0<T\le \infty $$ , $$1<p<N$$ and $$q<p$$ . We assume that A(t, x) is a coercive, bounded and measurable matrix, the growth rate q of the gradient term is superlinear but still subnatural, $$\gamma $$ is a positive constant, and the initial datum $$u_0$$ is an unbounded function belonging to a well precise Lebesgue space $$L^\sigma (\Omega )$$ for $$\sigma =\sigma (q,p,N)$$ .

Second-order two-scale analysis for heating effect of periodical composite structure

The multi-scale analysis method is presented for the heat transfer problems of periodical composite structures under irradiation heating. This kind of physical problem can be described as a class of parabolic equations with small periodical oscillating coefficients equipped with the mixed boundary conditions of both Dirichlet and Neumann type. It needs more expensive cost in both computer's memory and CPU time to solve these problems by the usual finite difference method or finite element method, since it is necessary to partition the considering geometrical region finer to capture the heat conduction behavior at small scale. To reduce the computational complexity, an approximate solution is derived by the asymptotic expansion technique. The convergence of asymptotic solution to exact solution is technically proved as the micro-scale parameter tends to zero. Based on the asymptotic expansion formula, a second-order two-scale finite element method is proposed to fully solve this problem in practice. Numerical results show convergent behavior of the proposed method.

Existence of Positive Solutions and Asymptotic Behavior for Evolutionary q(x)-Laplacian Equations

In this paper, we extend the variational method of M. Agueh to a large class of parabolic equations involving q(x)-Laplacian parabolic equation ∂ρt,x/∂t=divxρt,x∇xG′ρ+Vqx−2∇xG′ρ+V. The potential V is not necessarily smooth but belongs to a Sobolev space W1,∞Ω. Given the initial datum ρ0 as a probability density on Ω, we use a descent algorithm in the probability space to discretize the q(x)-Laplacian parabolic equation in time. Then, we use compact embedding W1,q.Ω↪↪Lq.Ω established by Fan and Zhao to study the convergence of our algorithm to a weak solution of the q(x)-Laplacian parabolic equation. Finally, we establish the convergence of solutions of the q(x)-Laplacian parabolic equation to equilibrium in the p(.)-variable exponent Wasserstein space.

On parabolic equations with critical electromagnetic potentials

We consider a class of parabolic equations with critical electromagnetic potentials, for which we obtain a classification of local asymptotics, unique continuation results, and an integral representation formula for solutions.

Coercive Solvability of Nonlocal Boundary-Value Problems for Parabolic Equations

In an arbitrary Banach space E, we consider the nonlocal problem $$ {\displaystyle \begin{array}{l}\upsilon^{\prime }(t)+A(t)\upsilon (t)=f(t)\kern1em \left(0\le t\le 1\right),\\ {}\upsilon (0)=\upsilon \left(\uplambda \right)+\mu \kern1em \left(0<\uplambda \le 1\right)\end{array}} $$ for an abstract parabolic equation with a linear unbounded strongly positive operator A(t) such that its domain D = D(A(t)) is independent of t and is everywhere dense in E. This operator generates an analytic semigroup exp{−sA(t)}(s ≥ 0). We prove the coercive solvability of the problem in the Banach space $$ {C}_0^{\alpha, \alpha}\left(\left[0,1\right],E\right)\left(0<\alpha <1\right) $$ with weight (t + τ)α. Earlier, this result was known only for constant operators. We consider applications in the class of parabolic functional differential equations with transformation of spatial variables and in the class of parabolic equations with nonlocal conditions on the boundary of the domain. Thus, this describes parabolic equations with nonlocal conditions both with respect to time and with respect to spatial variables.

The maximum norm analysis of a nonmatching grids method for a class of parabolic equation with nonlinear source terms

Motivated by the idea which has been introduced by M. Haiour and S.Boulaaras (Proc. Indian Acad. Sci. (Math. Sci.) Vol. 121,No. 4, November 2011,pp.481--493), We provide a maximum norm analysis of Euler scheme combined with finite element Schwarz alternating method for a class of parabolic equation with nonlinear source terms on two overlapping subdomains with nonmatching grids. We consider a domain which is the union of two overlapping subdomains where each subdomain has its own independently generated grid. The two meshes being mutually independent on the overlap region, a triangle belonging to one triangulation does not necessarily belong to the other one. Under a stability analysis on the theta scheme which given by our work in (App. Math. Comp., 217, 6443--6450 (2011).), we establish, on each subdomain, an optimal asymptotic behavior between the discrete Schwarz sequence and the asymptotic solution of parabolic differential equations.

Logarithmic upper bounds for weak solutions to a class of parabolic equations

AbstractIt is well known that a weak solution φ to the initial boundary value problem for the uniformly parabolic equation $\partial _t\varphi - {\rm div}(A\nabla \varphi ) +\omega \varphi = f $ in $\Omega _T\equiv \Omega \times (0,T)$ satisfies the uniform estimate $$\Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi\Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{q,\Omega_T}, \ \ \ c=c(N,\lambda, q, \Omega_T), $$ provided that $q \gt 1+{N}/{2}$, where Ω is a bounded domain in ${\open R}^N$ with Lipschitz boundary, T &gt; 0, $\partial _p\Omega _T$ is the parabolic boundary of $\Omega _T$, $\omega \in L^1(\Omega _T)$ with $\omega \ges 0$, and λ is the smallest eigenvalue of the coefficient matrix A. This estimate is sharp in the sense that it generally fails if $q=1+{N}/{2}$. In this paper, we show that the linear growth of the upper bound in $\Vert f \Vert_{q,\Omega _T}$ can be improved. To be precise, we establish $$ \Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi_0 \Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{1+{N}/{2},\Omega_T} \left(\ln(\Vert f \Vert_{q,\Omega_T}+1)+1\right). $$

Asymptotic behavior for a class of parabolic equations in weighted variable Sobolev spaces

Asymptotic behavior for a class of parabolic equations in weighted variable Sobolev spaces

Integral equation of Toeplitz plus Hankel's type and parabolic equation related to the Kontorovich‐Lebedev‐Fourier generalized convolutions

In this paper, we consider a class of integral equations with two Toeplitz plus Hankel's type kernels, and a class of parabolic equations. Using the Kontorovich‐Lebedev transform method, we obtain the closed form of solutions related to the Kontorovich‐Lebedev‐Fourier generalized convolutions. Furthermore, the estimate of solution in Lp with weight is established.

A Class of Parabolic Equations Driven by the Mean Curvature Flow

AbstractWe study a class of parabolic equations which can be viewed as a generalized mean curvature flow acting on cylindrically symmetric surfaces with a Dirichlet condition on the boundary. We prove the existence of a unique solution by means of an approximation scheme. We also develop the theory of asymptotic stability for solutions of general parabolic problems.

On the time evolution of Bernstein processes associated with a class of parabolic equations

In this article dedicated to the memory of Igor D. Chueshov, I first summarize in a few words the joint results that we obtained over a period of six years regarding the long-time behavior of solutions to a class of semilinear stochastic parabolic partial differential equations. Then, as the beautiful interplay between partial differential equations and probability theory always was close to Igor's heart, I present some new results concerning the time evolution of certain Markovian Bernstein processes naturally associated with a class of deterministic linear parabolic partial differential equations. Particular instances of such processes are certain conditioned Ornstein-Uhlenbeck processes, generalizations of Bernstein bridges and Bernstein loops, whose laws may evolve in space in a non trivial way. Specifically, I examine in detail the time development of the probability of finding such processes within two-dimensional geometric shapes exhibiting spherical symmetry. I also define a Faedo-Galerkin scheme whose ultimate goal is to allow approximate computations with controlled error terms of the various probability distributions involved.

Optimal control of parabolic equations by spectral decomposition

This paper deals with an optimal control problem of Bolza type for a class of parabolic equations. It consists in finding the initial datum that minimizes a cost functional, which comprises an energy term on the control and a regulation term given by a distributed cost on the state, and such that the final state lies within a prescribed distance to a given target. Beyond the particularities of the chosen problem, our aim here is to propose a new methodology based on a spectral decomposition of the operator governing the evolution of the system, describe its implementation for this kind of problem, and finally perform some preliminary numerical experiments to assess its behaviour.

Uniqueness of renormalized solutions for a class of parabolic equations

In this paper, we prove uniqueness of renormalized solution for a class of doubly nonlinear parabolic problems. $$\begin{aligned} \left\{ \begin{array}{lll}\displaystyle \frac{\partial e^{\beta u}}{\partial t} - \triangle _{p}u + {\text {div}}(c(x,t)|u|^{\gamma -1}u) + d(x,t)|\nabla u|^{\delta -1} = f -{\text {div}}(F) &{} {\text { in }}\, Q_{T},\\ u(x,t)=0 &{} {\text { on }} \partial \varOmega \times (0,T),\\ e^{\beta (u(x,0))}=e^{\beta (u_{0}(x))} &{} {\text { in }} \varOmega . \end{array} \right. \end{aligned}$$ (1)

Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations

AbstractIn this paper we develop explicit fast exponential Runge-Kutta methods for the numerical solutions of a class of parabolic equations. By incorporating the linear splitting technique into the explicit exponential Runge-Kutta schemes, we are able to greatly improve the numerical stability. The proposed numerical methods could be fast implemented through use of decompositions of compact spatial difference operators on a regular mesh together with discrete fast Fourier transform techniques. The exponential Runge-Kutta schemes are easy to be adopted in adaptive temporal approximations with variable time step sizes, as well as applied to stiff nonlinearity and boundary conditions of different types. Linear stabilities of the proposed schemes and their comparison with other schemes are presented. We also numerically demonstrate accuracy, stability and robustness of the proposed method through some typical model problems.

Numerical analysis of parabolic problems with dynamic boundary conditions

Space and time discretizations of parabolic differential equations with dynamic boundary conditions are studied in a weak formulation that fits into the standard abstract formulation of parabolic problems, just that the usual L^2(\Omega) inner product is replaced by an L^2(\Omega) \oplus L^2(\Gamma) inner product. The class of parabolic equations considered includes linear problems with time- and space-dependent coefficients and semilinear problems such as reaction-diffusion on a surface coupled to diffusion in the bulk. The spatial discretization by finite elements is studied in the proposed framework, with particular attention to the error analysis of the Ritz map for the elliptic bilinear form in relation to the inner product, both of which contain boundary integrals. The error analysis is done for both polygonal and smooth domains. We further consider mass lumping, which enables us to use exponential integrators and bulk-surface splitting for time integration.

Existence and qualitative properties of a solution of the first mixed problem for a parabolic equation with non-power-law double nonlinearity

The first mixed problem is investigated for a certain class of parabolic equations with double non-power-law nonlinearities in a cylindrical domain of the form . The domain can be unbounded. The existence of strong solutions in a Sobolev-Orlicz space is proved by the method of Galerkin approximations. A maximum principle is established, and upper and lower bounds characterizing the power-law decay of solution as are proved. The uniqueness of the solution is proved under certain additional assumptions. Bibliography: 29 titles.

Single-Mode and Dual-Mode Nongomogeneous Dissipative Structures in the Nonlocal Model of Erosion

We consider a periodic boundary-value problem for a nonlinear equation with the deviating spatial argument in the case when the deviation is small. This equation is called a spatially nonlocal erosion equation. It describes the formation of undulating surface relief under the influence of ion bombardment and can be interpreted as a development of the well-known Bradley-Harper model. It is shown that the nonhomogeneous surface relief can occur when the stability of the homogeneous states of equilibrium changes. In this boundary value problem the loss of stability can occur at the higher modes and a number of such modes. The mode number depends on many factors. For example, it depends on the angle of incidence. It is also shown that the nonlinear boundary value problem can be included into the class of abstract parabolic equations. Solvability of this problem was studied in the works by P.E. Sobolevsky, and this method assumes to use the analytical theory of semigroups of bounded linear operators. In order to solve the occurring bifurcation problems there were used the investigation methods of dynamical systems with an infinite-dimensional phase space (a space of initial conditions) such as: the method of integral manifolds, the method of Poincare–Dulac normal forms and asymptotic methods of analysis. Both possible in the given situation problems were studied: in codimension one and in codimension two. In particular, asymptotic formulas were obtained for solutions which describe nonhomogeneous undulating surface relief. The question about the stability of these solutions was studied. And the analysis of normal form was given. Also the asymptotic formulas for the nonhomogeneous undulating solutions were obtained. In conclusion some possible interpretations of the obtained results are indicated.

Long-Time Behavior of Solutions to a Class of Parabolic Equations with Nonstandard Growth Condition

In this paper, the author studies the long-time behavior of solutions to the p(x)-Laplace equation \(u_t=\mathrm {div}(|\nabla u|^{p(x)-2}\nabla u)+f(u)\), with homogeneous Dirichlet boundary condition in a bounded domain. As for the blow-up results, it is shown, by using the energy method, that the solutions of this problem blow up in finite time for nonpositive initial energy, or even for small positive initial energy. As for the extinction results, we give some sufficient conditions for the solutions to vanish in finite time. All these results generalize the ones when p(x) is a constant.