A semilinear parabolic problem with singular term at the boundary

In this paper, we deal with a class of semilinear parabolic problems related to a Hardy inequality with singular weight at the boundary.

More precisely, we consider the problem

$$\left\{\begin{array}{l@{\quad}l}u_t-\Delta u=\lambda \frac{u^p}{d^2}&\text{ in }\,\Omega_{T}\equiv\Omega \times (0,T), \\u>0 &\text{ in }\,{\Omega_T}, \\u(x,0)=u_0(x)>0 &\text{ in }\,\Omega, \\u=0 &\text{ on }\partial \Omega \times (0,T),\end{array}\right.$$

(P)

where Ω is a bounded regular domain of \({\mathbbm{R}^N}\), \({d(x)=\text{dist}(x,\partial\Omega)}\), \({p > 0}\), and \({\lambda > 0}\) is a positive constant.

We prove that

1. 1.
   If \({0 < p < 1}\), then (P) has no positive very weak solution.
    
2. 2.
   If \({p=1}\), then (P) has a positive very weak solution under additional hypotheses on \({\lambda}\) and \({u_0}\).
    
3. 3.
   If \({p > 1}\), then, for all \({\lambda > 0}\), the problem (P) has a positive very weak solution under suitable hypothesis on \({u_0}\).
    

Moreover, we consider also the concave–convex-related case.

     

A nonlinear parabolic equation with a nonlocal boundary term

A nonlinear parabolic problem with a nonlocal boundary condition is studied. We prove the existence of a solution for a monotonically increasing and Lipschitz continuous nonlinearity. The approximation method is based on Rothe’s method. The solution on each time step is obtained by iterations, convergence of which is shown using a fixed-point argument. The space discretization relies on FEM. Theoretical results are supported by numerical experiments.     

Partial reconstruction of the source term in a linear parabolic initial problem with first order boundary conditions

We consider the inverse problem of identifying a factor in the source term of a parabolic mixed system with first-order boundary conditions. The supplementary information, which is needed to do it, is given by the knowledge of the evolution in time of a certain integral with respect to a suitable Borel measure.     

On the boundary controllability of non-scalar parabolic systems

This Note is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional linear parabolic equations are considered. We show that, with boundary controls, the situation is much more complex than for similar distributed control systems. In our main result, we provide necessary and sufficient conditions for null controllability. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Controlling the moving boundary of the parabolic obstacle problem

We consider the controlled parabolic obstacle problem with the controlu appearing through the Neumann condition y/v=u on the boundary of an open domain ofR ^N . Given a smooth open subsetQ ⁰ of ×(0,T) we prove that the problemQ ⁰ {(x, t) Q, y ^u (x, t)=0} is approximately solvable. Numerical tests are given.This work was done while Viorel Barbu was the Otto Szasz Visiting Professor at the University of Cincinnati.     

Reconstruction of boundary controls in parabolic systems

In the paper, an inverse dynamic problem is considered. It consists in reconstructing a priori unknown boundary controls in dynamical systems described by boundary value problems for partial differential equations of parabolic type. The source information for solving the inverse problem is the results of approximate measurements of the states of the observed system’s motion. The problem is solved in the static case; i.e., to solve it, we use all the measurement data accumulated during some specified observation interval. The problem under consideration is ill-posed. To solve it, we propose the Tikhonov method with a stabilizer containing the sum of the mean-square norm and total time variation of the control. The use of such nondifferentiable stabilizer allows us to obtain more precise results than the approximation of the desired control in the Lebesgue spaces. In particular, this method provides the pointwise and piecewise uniform convergences of regularized approximations and makes possible the numerical reconstruction of the subtle structure of the desired control. In the paper, the subgradient projection method for obtaining a minimizing sequence for the Tikhonov functional is described and substantiated. Also, we demonstrate the two-stage finitedimensional approximation of the problem and present the results of numerical simulation.     

Blow-up of solutions of a semilinear hyperbolic equation and a parabolic equation with general forcing term and boundary condition

Semilinear hyperbolic and parabolic initial–boundary value problems are studied. Criteria for solutions of a semilinear hyperbolic equation and a parabolic equation with general forcing term and general boundary condition to blow up in finite time are obtained.     

Well-posedness and regularity of boundary feedback parabolic systems

A general second order parabolic equation is considered with both Dirichlet or mixed (in particular, Neumann) input function acting on the boundary S of the bounded spatial domain Ω. The distinctive new feature is that the input function is demanded to be expressed in feedback form, i.e. as a linear operator (of finite dimensional range) of the solution, continuous from $\text{H}{}^{\mathrm{s}}\text{(Ω)}$ into L_p(S), for some non-negative real s and for p ? 1. Well posedness and regularity results of the resulting closed loop system are established in appropriate functions spaces. The results are illustrated by examples of physical interest.     

The boundary regularity of non-linear parabolic systems II

This is the second part of a work aimed at establishing that for solutions to Cauchy–Dirichlet problems involving general non-linear systems of parabolic type, almost every parabolic boundary point is a Hölder continuity point for the spatial gradient of solutions. Here we establish higher fractional differentiability of solutions up to the boundary. Based on the necessary and sufficient condition for regular boundary points from the first part of Bögelein et al. (in this issue)[7] we achieve dimension estimates for the boundary singular set and eventually the almost everywhere regularity of solutions at the boundary.     

Constructing eigenfunctions of non-selfadjoint coupled parabolic boundary problems

This paper deals with the eigenfunction construction of coupled parabolic boundary value problems under a more general situation than in previous papers where a certain matrix related to the boundary conditions must have real eigenvalues.     

The boundary regularity of non-linear parabolic systems I

This is the first part of a work aimed at establishing that for solutions to Cauchy–Dirichlet problems involving general non-linear systems of parabolic type, almost every parabolic boundary point is a Hölder continuity point for the spatial gradient of solutions. Here we develop the basic necessary and sufficient condition for establishing the regular nature of a boundary point.     

Higher regularity of the free boundary in the parabolic Signorini problem

We show that the quotient of two caloric functions which vanish on a portion of an \(H^{k+ \alpha }\) regular slit is \(H^{k+ \alpha }\) at the slit, for \(k \ge 2\). In the case \(k=1\), we show that the quotient is in \(H^{1+\alpha }\) if the slit is assumed to be space-time \(C^{1, \alpha }\) regular. This can be thought of as a parabolic analogue of a recent important result in De Silva and Savin (Boundary Harnack estimates in slit domains and applications to thin free boundary problems, 2014), whose ideas inspired us. As an application, we show that the free boundary near a regular point of the parabolic thin obstacle problem studied in Danielli et al. (Optimal regularity and the free boundary in the parabolic Signorini problem. Mem. Am. Math. Soc., 2013) with zero obstacle is \(C^{\infty }\) regular in space and time.