Class Of Singular Parabolic Equations Research Articles

Global Existence and Blow-up of Solutions for a Class of Singular Parabolic Equations with Viscoelastic Term

In this paper, we consider the initial boundary value problem for a class of singular parabolic equations with viscoelastic term and logarithmic term. By using the technique of cut-off and the method of Faedo-Galerkin approximation, the local existence of the weak solution is established. Based on the potential well method, the global existence of the weak solution is derived. Furthermore, we prove that the weak solution blows up in finite time by taking the concavity analysis method.

Carleman estimates and null controllability of a class of singular parabolic equations

Abstract In this paper, we consider control systems governed by a class of semilinear parabolic equations, which are singular at the boundary and possess singular convection and reaction terms. The systems are shown to be null controllable by establishing Carleman estimates, observability inequalities and energy estimates for solutions to linearized equations.

Backward estimates for nonnegative solutions to a class of singular parabolic equations

In this paper we improve a recent result of the same Authors (Recalde and Vespri, 2015) by using the same approach in a more tricky way. More precisely, we study a class of quasilinear singular parabolic equations with L∞ coefficients, whose prototypes are the p-Laplacian (2NN+1<p<2) equations. Let us consider nonnegative solutions defined in R+×RN. We recall that in Recalde and Vespri (2015), starting from the value attained in a point (x0,t0) by the solution, we proved sharp estimates in the stripe ((1−ε)t0,∞)×RN. In this note we show that, quite surprisingly, sharp estimates hold also for the remote past i.e. also for the stripe (0,t0)×RN.In the last section we briefly show how these results can be adapted to equations of Porous Medium type in the fast diffusion range i.e. (N−2N)+<m<1.

Harnack estimates at large: Sharp pointwise estimates for nonnegative solutions to a class of singular parabolic equations

In this paper we deal with quasilinear singular parabolic equations with L∞ coefficients, whose prototypes are the p-Laplacian (2NN+1<p<2) equations. In this range of the parameters, we are in the so called fast diffusion case. Extending a recent result (Ragnedda et al. 2013), we are able to prove Harnack estimates at large, i.e. starting from the value attained in a point by the solution, we are able to give explicit and sharp pointwise estimates, from below by using the Barenblatt solutions. In the last section we briefly show how these results can be adapted to equations of porous medium type in the fast diffusion range i.e. (N−2N)+<m<1.

Conservation of the mass for solutions to a class of singular parabolic equations

In this paper we deal with the Cauchy problem associated to a class of quasilinear singular parabolic equations with L∞ coefficients, whose prototypes are the p-Laplacian $\left(\frac{2N}{N+1}<p<2\right)$ and the Porous medium equation $\left(\left(\frac{N-2}{N}\right)_+<m<1\right)$. In this range of the parameters p and m, we are in the so called fast diffusion case. We prove that the initial mass is preserved for all the times.

Existence and asymptotic behavior of solutions to a singular parabolic equation

In this paper, we study the initial-boundary value problem for a class of singular parabolic equations. Under some conditions, we obtain the existence and asymptotic behavior of solutions to the problem by parabolic regularization method and the sub-super solutions method. As a byproduct, we prove the existence of solutions to some problems with gradient terms, which blow up on the boundary.

Harnack estimates for non-negative weak solutions of a class of singular parabolic equations

We prove forward, backward and elliptic Harnack type inequalities for non-negative local weak solutions of singular parabolic differential equations of type $$u_t={\rm div}{\bf A}(x, t, u, Du)$$ where A satisfies suitable structure conditions and a monotonicity assumption. The prototype is the parabolic p−Laplacian with 1 < p < 2. By using only the structure of the equation and the comparison principle, we generalize to a larger class of equations the estimates first proved by Bonforte et al. (Adv. Math. 224, 2151–2215, 2010) for the model equation.

Existence of the maximal weak solution for a class of singular parabolic equations

In this paper, we study the initial–boundary value problem for a class of singular parabolic equations. Under some additional conditions, we prove that there exists a weak solution for the problem by parabolic regularization. Moreover, we show that the solution obtained by parabolic regularization is the maximal solution among all weak solutions.

On three‐point boundary value problem with a weighted integral condition for a class of singular parabolic equations

We deal with a three point boundary value problem for a class of singular parabolic equations with a weighted integral condition in place of one of standard boundary conditions. We will first establish an a priori estimate in weighted spaces. Then, we prove the existence, uniqueness, and continuous dependence of a strong solution.

On the solvability of a class of singular parabolic equations with nonlocal boundary conditions in nonclassical function spaces

The aim of this paper is to prove the existence, uniqueness, and continuous dependence upon the data of a generalized solution for certain singular parabolic equations with initial and nonlocal boundary conditions. The proof is based on an a priori estimate established in nonclassical function spaces, and on the density of the range of the operator corresponding to the abstract formulation of the considered problem.

A boundary modulus of continuity for a class of singular parabolic equations

Parabolic equations describing diffusion phenomena with change of phase are considered. It is demonstrated that weak solutions are continuous up to the parabolic boundary of the domain of definition. The continuity is quantitatively described by a modulus determined a priori only in terms of the data.

On the numerical methods for the singular parabolic equations in fluid dynamics

The concern here is with the numerical methods for the well-known singular parabolic equations in unsteady boundary-layer flows behind a moving shock and the development of boundary-layers on a semi-infinite plate impulsively set into motion. These equations can be derived with a semisimilar transformation by which the domains of solution are mapped on to [0, 1]. It is shown that this class of singular parabolic equations can be elegantly and accurately solved by using the successive-overrelaxation method normally applied to elliptic equations. No artificial viscosity or numerical attenuation is required. Details of the computational procedures are given. Salient natures of the numerical method are analyzed. Results of the analysis show that as far as the stability of computations is concerned, the sign of the diffusivity is not important; however, to obtain convergent solution, the “initial” conditions have to be specified consistently.

A Singular Parabolic Initial-Boundary Value Problem in a Noncylindrical Domain

The well-posedness of the first Fourier problem for a class of singular parabolic equations in noncylindrical domain is established in an appropriate weighted Hilbert–Sobolev space. The Green’s function is also constructed by means of potentials, and it is shown that the generalized and classical solutions coincide when the data permit the latter to exist.

Singular perturbations for a singular parabolic initial boundary value problem

The first Fourier problem is considered for a class of singular parabolic equations with a small parameter multiplying the time derivative term. Boundary layer corrections for t close to t=0 are constructed by a singular perturbation procedure and the asymptotic correctness of the approximation is established.