Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators

Our first basic model is the fully nonlinear dual porous medium equation with source

Existence for a degenerate Cauchy problem

We prove the existence of a solution to the degenerate parabolic Cauchy problem with a possibly unbounded Radon measure as an initial data. To accomplish this, we establish a priori estimates and derive a compactness result. We also show that the result is optimal in the Euclidian setting.     

Exponential attractors for a class of reaction-diffusion problems with time delays

We consider a reaction-diffusion system subject to homogeneous Neumann boundary conditions on a given bounded domain. The reaction term depends on the population densities as well as on their past histories in a very general way. This class of systems is widely used in population dynamics modelling. Due to its generality, the longtime behavior of the solutions can display a certain complexity. Here we prove a qualitative result which can be considered as a common denominator of a large family of specific models. More precisely, we demonstrate the existence of an exponential attractor, provided that a bounded invariant region exists and the past history decays exponentially fast. This result will be achieved by means of a suitable adaptation of the l-trajectory method coming back to the seminal paper of Málek and Nečas. The first author was partially supported by the Italian PRIN 2006 research project Problemi a frontiera libera, transizioni di fase e modelli di isteresi. The second author was supported by the research project MŠM 0021620839 and by the project LC06052 (Jindřich Nečas Center for Mathematical Modeling).     

Local estimates and global existence for strongly nonlinear parabolic equations with locally integrable data

We obtain existence results for some strongly nonlinear Cauchy problems posed in and having merely locally integrable data. The equations we deal with have as principal part a bounded, coercive and pseudomonotone operator of Leray-Lions type acting on , they contain absorbing zero order terms and possibly include first order terms with natural growth. For any p > 1 and under optimal growth conditions on the zero order terms, we derive suitable local a-priori estimates and consequent global existence results.     

Regularity results for quasilinear elliptic equations in the Heisenberg group

We prove regularity results for solutions to a class of quasilinear elliptic equations in divergence form in the Heisenberg group . The model case is the non-degenerate p-Laplacean operator where , and p is not too far from 2.     

Non-autonomous semilinear evolution equations with almost sectorial operators

Inspired by the theory of semigroups of growth α, we construct an evolution process of growth α. The abstract theory is applied to study semilinear singular non-autonomous parabolic problems. We prove that, under natural assumptions, a reasonable concept of solution can be given to such semilinear singularly non-autonomous problems. Applications are considered to non-autonomous parabolic problems in space of H?lder continuous functions and to a parabolic problem in a domain with a one dimensional handle. Marcelo J. D. Nascimento: Research partially supported by FAPESP # 02/11855-2, Brazil.     

Stability of the difference type methods for linear Volterra equations in Hilbert spaces

We study the stability of backward difference timestepping method for linear Volterra equations of scalar type in a Hilbert space framework. The results and methods extend and simulate numerically those introduced by Prüss for integrability with respect to continuous solutions. This work was supported in part by the National Natural Science Foundation of China, contract grant number 10271046.     

Kolmogorov equations for measures

We consider a semigroup of operators in the Banach space C _b (H) of uniformly continuous and bounded functions on a separable Hilbert space H. We prove an existence and uniqueness result for a measure valued equation involving this class of semigroups. Then we apply the result to the transition semigroup and the Kolmogorov operator corresponding to a stochastic PDE in H. For this purpose, we characterize the generator of the transition semigroup on a core.      

Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian

The asymptotic behavior of viscosity solutions to the Cauchy–Dirichlet problem for the degenerate parabolic equation u _t  = Δ_∞ u in Ω × (0,∞), where Δ_∞ stands for the so-called infinity-Laplacian, is studied in three cases: (i) and the initial data has a compact support; (ii) Ω is bounded and the boundary condition is zero; (iii) Ω is bounded and the boundary condition is non-zero. Our method of proof is based on the comparison principle and barrier function arguments. Explicit representations of separable type and self-similar type of solutions are also established. Moreover, in case (iii), we propose another type of barrier function deeply related to a solution of . Goro Akagi was supported by the Shibaura Institute of Technology grant for Project Research (no. 2006-211459, 2007-211455), and the grant-in-aid for young scientists (B) (no. 19740073), Ministry of Education, Culture, Sports, Science and Technology. Petri Juutinen was supported by the Academy of Finland project 108374. Ryuji Kajikiya was supported by the grant-in-aid for scientific research (C) (no. 16540179), Ministry of Education, Culture, Sports, Science and Technology.     

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

We study the limit behaviour of solutions of with initial data k δ ₀ when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r ^β , β > N(p − 1) − 2, we prove that the limit function u _∞ is an explicit very singular solution, while such a solution does not exist if β ≤  N(p − 1) − 2. If lim inf _{r→ 0} r ² ln (1/h(r))  >  0, u _∞ has a persistent singularity at (0, t) (t ≥  0). If , u _∞ has a pointwise singularity localized at (0, 0).     

Existence theorems for nonlinear functional differential equations of neutral type

Conditions are found upon satisfaction of which the differential equation

Kernel estimates for a class of Kolmogorov semigroups

We prove short time pointwise upper bounds for the heat kernels of certain Kolmogorov operators. We use Lyapunov function techniques, where the Lyapunov functions depend also on the time variable. Received: 29 November 2007     

A free-boundary problem for the evolution <Emphasis Type="Italic">p</Emphasis>-Laplacian equation with a combustion boundary condition

We study the existence, uniqueness and regularity of solutions of the equation f _t  = Δ _p f = div (|Df| ^p-2 Df) under over-determined boundary conditions f = 0 and |Df| = 1. We show that if the initial data is concave and Lipschitz with a bounded and convex support, then the problem admits a unique solution which exists until vanishing identically. Furthermore, the free-boundary of the support of f is smooth for all positive time.     

Existence of Periodic Solutions for the Quasi-Static Thermoelastic Thermistor Problem

We deal with a new model for the thermistor problem formulated as a coupled system of PDE’s involving nonlinear energy heat equation, stationary charge conservation equation of electrical current and thermoelastic equations of displacement. We establish the existence of weak periodic solutions rewriting our system as an abstract problem in order to utilize the maximal monotone mappings theory and a fixed point argument for a suitable operator equation.      

A new iterative method for discrete HJB equations

A successive relaxation iterative algorithm for discrete HJB equations is proposed. Monotone convergence has been proved for the algorithm. This work was supported by NNSF of China (no. 10571046).     

Remarks on extremal solutions of differential equations with advanced argument

We show the existence of absolutely continuous extremal solutions to the problemx′(t)=f(t, x)h(t)))+g(t)),x(0)=x ₀, whereh is an arbitrary continuous deviated argument. Conditions for the uniqueness of solutions are given. Research partialy supported by grant UG BW 5100 - 5 - 0143 - 4     

Existence of type (II) regions and convexity and concavity of potential functionals corresponding to jumping nonlinear problems

In this paper we study the jumping nonlinear problem

Regularity theory in Orlicz spaces for the parabolic polyharmonic equations

In this paper we generalize classical L ^p estimates to Orlicz spaces for the parabolic polyharmonic equations. Our argument is based on the iteration-covering procedure. Received: 10 September 2007     

Sign-changing stationary solutions and blowup for the nonlinear heat equation in a ball

In this paper, we prove that if is a radially symmetric, sign-changing stationary solution of the nonlinear heat equation

Feedback stabilization of a class of evolution equations with delay

In this paper, we characterize the stabilization of some delay systems. The proof of the main result uses the method introduced in Ammari and Tucsnak (ESAIM COCV 6:361–386, 2001) where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined with a boundedness property of the transfer function of the associated open loop system. The author would like to thank the Institute of Research for Development of France (IRD) and LMDP, the Laboratory of Mathematics and Dynamic of Populations in Marrakesh, for supporting his visit.