On the existence of solutions to a special variational problem

In this paper we establish an existence and regularity result for solutions to the problem

Local regularity of weak solutions of semilinear parabolic systems with critical growth

We show that, under so called controllable growth conditions, any weak solution in the energy class of the semilinear parabolic system

On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source

This paper is concerned with the study of the nonlinear damped wave equation

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.     

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).     

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).     

Mathematical and numerical study of a system of conservation laws

The system of equations (f (u))_t − (a(u)v + b(u))_x = 0 and u_t − (c(u)v + d(u))_x = 0, where the unknowns u and v are functions depending on , arises within the study of some physical model of the flow of miscible fluids in a porous medium. We give a definition for a weak entropy solution (u, v), inspired by the Liu condition for admissible shocks and by Krushkov entropy pairs. We then prove, in the case of a natural generalization of the Riemann problem, the existence of a weak entropy solution only depending on x/t. This property results from the proof of the existence, by passing to the limit on some approximations, of a function g such that u is the classical entropy solution of u_t − ((cg + d)(u))_x = 0 and simultaneously w = f (u) is the entropy solution of w_t − ((ag + b)(f⁽⁻¹⁾(w)))_x = 0. We then take v = g(u), and the proof that (u, v) is a weak entropy solution of the coupled problem follows from a linear combination of the weak entropy inequalities satisfied by u and f (u). We then show the existence of an entropy weak solution for a general class of data, thanks to the convergence proof of a coupled finite volume scheme. The principle of this scheme is to compute the Godunov numerical flux with some interface functions ensuring the symmetry of the finite volume scheme with respect to both conservation equations.     

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

Far field asymptotics of solutions to convection equation with anomalous diffusion

The initial value problem for the conservation law is studied for and under natural polynomial growth conditions imposed on the nonlinearity. We find the asymptotic expansion as of solutions to this equation corresponding to initial conditions, decaying sufficiently fast at infinity. The preparation of this paper was supported in part by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389, and in part by the Polonium Project PAI EGIDE N. 09361TG. The first author gratefully thanks the Mathematical Institut of Wrocław University for the warm hospitality. The preparation of this paper by the second author was also partially supported by the grant N201 022 32 / 09 02.     

Nonuniqueness in the quenching problem

This paper deals with nonnegative solutions of for

Non-simple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation

For the Neumann sinh-Gordon equation on the unit ball

Elliptic estimates independent of domain expansion

In this paper, we consider elliptic estimates for a system with smooth variable coefficients on a domain containing the origin. We first show the invariance of the estimates under a domain expansion defined by the scale that with parameter R > 1, provided that the coefficients are in a homogeneous Sobolev space. Then we apply these invariant estimates to the global existence of unique strong solutions to a parabolic system defined on an unbounded domain. This paper was supported in part by research funds of Chonbuk National University in 2007.     

Existence results for a class of porous medium type equations with a quadratic gradient term

In this paper we study the Dirichlet problem in Q _T = Ω × (0, T) for degenerate equations of porous medium-type with a lower order term:

On the long time behaviour of solutions to dissipative wave equations in \mathbb{R}^{2}

In this paper we present a parabolic approach to studying the diffusive long time behaviour of solutions to the Cauchy problem:

Local Well-posedness of Kinetic Chemotaxis Models

We present a general functional analytic setting in which the Cauchy problem for mild solutions of kinetic chemotaxis models is well-posed, locally in time, in general physical dimensions. The models consist of a hyperbolic transport equation that is non-linearly and non-locally coupled to a reaction-diffusion system through kernel operators. Three examples are elaborated throughout the paper in which the latter system is (1) a single linear equation, (2) a FitzHugh-Nagumo system and (3) a piecewise linear approximation thereof. Finally we present a limit argument to obtain results on solutions in L ¹ ∩ L ^∞. Sander C. Hille: This work is supported by PIONIER grant 600-61410 of the Netherlands Organisation for Scientific Research, NWO.     

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.     

Decay properties for the solutions of a partial differential equation with memory

Decay properties in energy norm for solutions of a class of partial differential equations with memory are studied by means of frequency domain methods. Our results are optimal for this class, as we are able to characterize polynomial as well as exponential decay rates. The results apply to models for viscoelastic materials. An extension to a semilinearly perturbed problem is also included. Received: 9 July 2008, Revised: 16 September 2008     

A note on the behavior of blow-up solutions for one-phase Stefan problems

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Optimal decay rates of solutions to dissipative nonlinear evolution equations with ellipticity

We derive the optimal decay rates of solution to the Cauchy problem for a set of nonlinear evolution equations with ellipticity and dissipative effects

Decay rates of solutions to dissipative nonlinear evolution equations with ellipticity

In this paper, we study the global existence and the asymptotic behavior of the solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects