We study the use of a GPU for the numerical approximation of the curvature dependent flows of graphs - the mean-curvature flow and the Willmore flow. Both problems are often applied in image processing where fast solvers are required. We approximate these problems using the complementary finite volume method combined with the method of lines. We obtain a system of ordinary differential equations which we solve by the Runge-Kutta-Merson solver. It is a robust solver with an automatic choice of the...

In this paper, the existence of an $\omega$-periodic weak solution of a parabolic equation (1.1) with the boundary conditions (1.2) and (1.3) is proved. The real functions $f(t,r),h\left(t\right),a\left(t\right)$ are assumed to be $\omega$-periodic in $t,f\in L_2(S,H),a,h$ such that $a^{\text{'}}\in L_{\infty }\left(R\right),h^{\text{'}}\in L_{\infty }\left(R\right)$ and they fulfil (3). The solution $u$ belongs to the space $L_2(S,V)\cap L_{\infty }(S,H)$, has the derivative $u^{\text{'}}\in L_2(S,H)$ and satisfies the equations (4.1) and (4.2). In the proof the Faedo-Galerkin method is employed.

In this paper the finite speed of propagation of solutions and the continuous dependence on the nonlinearity of a degenerate parabolic partial differential equation are discussed. Our objective is to derive an explicit expression for the speed of propagation and the large time behavior of the solution and to show that the solution continuously depends on the nonlinearity of the equation.

I shall discuss joint work with John L. Lewis on the solvability of boundary value problems for the heat equation in non-cylindrical (i.e., time-varying) domains, whose boundaries are in some sense minimally smooth in both space and time. The emphasis will be on the Neumann problem with data in $L^p$. A somewhat surprising feature of our results is that, in contrast to the cylindrical case, the optimal results hold when $p=2$, with the situation getting progressively worse as $p$ approaches $1$. In particular,...

For two general second order parabolic equations in divergence form in Lip(1,1/2) cylinders, we give a criterion for the preservation of $L^p$ solvability of the Dirichlet problems.

We consider the initial-boundary-value problem for the one-dimensional fast diffusion equation $u_t=[sign\left(u_x\right)log|u_x{\left|\right]}_x$ on $Q_T=[0,T]\times [0,l]$. For monotone initial data the existence of classical solutions is known. The case of non-monotone initial data is delicate since the equation is singular at $u_x=0$. We ‘explicitly’ construct infinitely many weak Lipschitz solutions to non-monotone initial data following an approach to the Perona-Malik equation. For this construction we rephrase the problem as a differential inclusion which enables us...

We give a new proof, based on analytic semigroup methods, of a maximal regularity result concerning the classical Cauchy-Dirichlet's boundary value problem for second order parabolic equations. More specifically, we find necessary and sufficient conditions on the data in order to have a strict solution $u$ which is bounded with values in $C^{2+\theta }\left(\overline{\mathrm{\Omega }}\right)$(0 < < 1), with${\partial }_{t}u$ bounded with values in $C^{\theta }\left(\overline{\mathrm{\Omega }}\right)$.

In this paper, we use the adapted periodic unfolding method to study the homogenization and corrector problems for the parabolic problem in a two-component composite with ε-periodic connected inclusions. The condition imposed on the interface is that the jump of the solution is proportional to the conormal derivative via a function of order εγ with γ ≤ −1. We give the homogenization results which include those obtained by Jose in [Rev. Roum. Math. Pures Appl. 54 (2009) 189–222]. We also get the...

We show that the study of the principal spectrum of a linear nonautonomous parabolic PDE of second order $u_t=\Delta u+a\left(t\right)u$ on a bounded domain, with the Dirichlet or Neumann boundary conditions, reduces to the investigation of the spectrum of the linear nonautonomous ODE v̇ = a(t)v.

We give necessary and sufficient conditions on the initial data such that the solutions of parabolic equations have a prescribed Sobolev regularity in time and space.