On parabolic systems with variational structure

We prove global existence of strong solutions for two-dimensional parabolic systems of variational flow type subject to the one-sided condition.     

Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure

We consider nonlinear parabolic systems of the form ut=−∇V(u)+u_xx$u_t=-\mathrm{\nabla }V(u)+u_{xx}$, where u∈Rn$u\in {\mathbb{R}}^n$, n?1$n?1$, x∈R$x\in \mathbb{R}$, and the potential V   is coercive at infinity. For such systems, we prove a result of global convergence toward bistable fronts which states that invasion of a stable homogeneous equilibrium (a local minimum of the potential) necessarily occurs via a traveling front connecting to another (lower) equilibrium. This provides, for instance, a generalization of the global convergence result obtained by Fife and McLeod [P. Fife, J.B. McLeod, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rat. Mech. Anal. 65 (1977) 335–361] in the case n=1$n=1$. The proof is based purely on energy methods, it does not make use of comparison principles, which do not hold any more when n>1$n>1$.     

On Toeplitz operators with quasihomogeneous symbols

In this paper, we give some basic results concerning Toeplitz operators whose symbol is of the form e^{i p θ}ϕ, where ϕ is a radial function, then use these results to characterize all Toeplitz operators which commute with them.Received: 12 June 2004; revised: 27 January 2005     

Quasilinear parabolic systems

Parabolic systems of partial differential equations are developed and applications are discussed. The systems are quasilinear in divergence form with high-order coefficient matrices which are neither symmetric nor sparse. Weak existence, uniqueness, and stability are established for an appropriate initial-boundary value problem.     

Regularity in parabolic systems with continuous coefficients

Under the only assumption of continuous coefficients, we prove a partial H?lder continuity result for solutions to parabolic systems with polynomial growth. A key component throughout the argument is the use of DiBenedetto’s intrinsic geometry (Degenerate Parabolic Equations. Universitext. Springer, New York, 1993) to accommodate the inhomogeneity in the system. A main technical point is that, although we are proving the H?lder continuity of solutions, we employ the intrinsic geometry using cylinder stretched according to the size of the (spatial) gradient, in an iteration scheme that does not necessarily imply the boundedness of the gradient itself.     

Cauchy problem for parabolic systems with pulse action

For linear parabolic systems with pulse action, we establish the well-posedness of the Cauchy problem in normed Dini spaces. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1525–1535, November, 2006.     

Optimal control for parabolic systems with competitive interactions

We consider a two-dimensional parabolic system with general competitive interactions as a two-player game with conflicting objectives and with controls on the inhomogencous (source) terms. We show the existence of an optimal solution of the game as the saddle point of a suitable objective functional.     

On global stability of parabolic systems with delay

A semilinear differential equation with delay in a Hilbert space is considered. Conditions for global stability are established. An application to parabolic systems with delay is discussed.     

Optimal control of parabolic systems with time-varying lags

Various optimal control problems for linear parabolic systemswith time-varying lags are considered. Necessary and sufficientconditions of optimality are derived for the Neumann problem.The optimal control is obtained in the feedback form. Makinguse of the results of Schwart's, the representation of the optimalfeedback control is given.     

Feedback control for parabolic systems with in-domain actuation

State feedback design for linear parabolic systems with in-domain actuation and general Robin boundary conditions is considered. To this end the system is shown to be state equivalent to a boundary controlled system. By means of the backstepping transformation the latter system is shown to be feedback equivalent to a stable parabolic equation. Within the contribution previous results concerning such systems are supplemented by numerical simulation studies. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Partial regularity for parabolic systems with non-standard growth

We study non-linear parabolic systems with non-standard p(z)-growth conditions and establish that the gradient of weak solutions is locally H?lder continuous with H?lder exponent b ? (0,1){\beta \in (0,1)} with respect to the parabolic metric on an open set of full Lebesgue measure, provided the exponent function p(z) itself is H?lder continuous with exponent β with respect to the parabolic metric.     

Blow-up rates for parabolic systems

Let ⁿ be a bounded domain andB _R be a ball in ⁿ of radiusR. We consider two parabolic systems: ut=u +f(), i= +g(u) in × (0,T) withu=v=0 on × (0,T) andu _t =u, v _t =v inB _r × (0,T) withe/v=f (v), e/v=g(u) onB _R × (0,T). Whenf(v) andg(u) are power law or exponential functions, we establish estimates on the blow-up rates for nonnegative solutions of the systems.     

Asymptotic expansions for parabolic systems

We consider a parabolic system in a half space. A theorem, similar to one proved by Meyers and Pazy for elliptic equations outside the unit ball is proved, namely, if the coefficients, the right side, and the initial conditions of the parabolic system have asymptotic expansions at infinity with respect to the space variable, then so does the solution of the corresponding Cauchy problem. Some generalizations and examples are given.