Nonlinear equations with unbounded heat conduction and integrable data

We consider a class of quasi-linear diffusion problems involving a matrix A(t,x,u) which blows up for a finite value m of the unknown u. Stationary and evolution equations are studied for L ¹ data. We focus on the case where the solution u can reach the value m. For such problems we introduce a notion of renormalized solutions and we prove the existence of such solutions.      

Nonlinear degenerate parabolic equations with measure data

In this paper, we study the existence of solution to nonlinear degenerate parabolic equations with measure data.     

Nonlinear anisotropic parabolic equations in with locally integrable data

In this paper, we prove existence and regularity of weak solutions for a class of nonlinear anisotropic parabolic problems in with locally integrable data. Our approach is based on the anisotropic Sobolev inequality, a smoothness, and compactness results. Copyright © 2012 John Wiley & Sons, Ltd.     

Nonlinear elliptic and parabolic equations

Results of recent years are presented on the theory of nonlinear elliptic and parabolic equations of any order including equations of infinite order.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 9, pp. 5–130, 1976.     

Nonlinear degenerate parabolic equations with Neumann boundary condition

We study Neumann problem for a class of nonlinear degenerate parabolic PDE. A typical nonlinearity we have in mind is, for instance, β(u)=−1/u(u>0). We establish a necessary and sufficient condition on given data for existence of solution.     

Degenerate parabolic equations with initial data measures

We address the problem of existence of solutions to degenerate (and nondegenerate) parabolic equations under optimal assumptions on the initial data, which are assumed to be measures. The requirements imposed on the initial data are connected both with the degeneracy of the principal part of the equation, and with the form of the nonlinear forcing term. The latter depends on the space gradient of a power of the solution. Applications to related problems are also outlined.

     

Nonlinear parabolic equations, relaxation and roughness

In a recent paper, Aubin and Coulouvrat (1998) dealt with equations of motion in the fluid in relaxation - mathematically: Burgers' equations, physically: continuity equations, Navier-Stokes equations, energy bilance, and equations of relaxation. A related equation of Ginzburg and Landau type (1965) extended for the kinetic depinning transitions takes its form

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Nonlinear parabolic equations, relaxation and roughness

In a recent paper, Aubin and Coulouvrat (1998) dealt with equations of motion in the fluid in relaxation - mathematically: Burgers' equations, physically: continuity equations, Navier-Stokes equations, energy bilance, and equations of relaxation. A related equation of Ginzburg-Landau type (1965) extended for the kinetic depinning transitions takes its form

(0)     

Doubly nonlinear parabolic equations with measure data

The existence of solutions to the initial boundary value problem for the equation $$u_{t}-{\rm div}(u^m|Du|^{p-2}Du)=\lambda|Du^q|^{l}+u^{\alpha},$$ with zero-Dirichlit boundary condition and Radon measure as initial condition is studied, where m > 0, p > 1, λ, q, l, and α in various situations.     

Global attractors for quasilinear parabolic equations on unbounded thin domains

Monatshefte für Mathematik - In this paper we are concerned with the asymptotic behavior of quasilinear parabolic equations posed in a family of unbounded domains that degenerates onto a lower...     

Nonlinear degenerate parabolic equations with time dependent singular coefficients

We are concerned with the nonexistence of positive solutions of the nonlinear parabolic partial differential equations in a cylinder Ω × (0, T) with initial condition u(., 0) = u₀(.) ? 0 and vanishing on the boundary ?Ω × (0, T), given by where $\Omega \in \mathbf {R}^NWe are concerned with the nonexistence of positive solutions of the nonlinear parabolic partial differential equations in a cylinder Ω × (0, T) with initial condition u(., 0) = u₀(.) ? 0 and vanishing on the boundary ?Ω × (0, T), given by where $\Omega \in \mathbf {R}^N$ (resp. a Carnot Carathéodory metric ball in $\mathbf {R}^{2N+1})$ with smooth boundary and the time dependent singular potential function V(x, t) ∈ L¹_loc(Ω × (0, T)), $\alpha , \beta \in \mathbf {R}$, 1 < p < N, p ? 1 + α + β > 0. We find the best lower bounds for p + β and provide proofs for the nonexistence of positive solutions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Nonlinear reaction diffusion problems with ultra parabolic limiting equations

In this paper the nonlinear reaction diffusion problems with ultraparabolic equa- tions are considered.By using comparison theorem,the existence,uniqueness and asymptotic behavior of solution for the problem are studied.     

Navier-stokes equations on unbounded domains with rough initial data

We consider the Navier-Stokes equations in unbounded domains Ω ⊆ ℝ ⁿ of uniform C ^1,1-type. We construct mild solutions for initial values in certain extrapolation spaces associated to the Stokes operator on these domains. Here we rely on recent results due to Farwig, Kozono and Sohr, the fact that the Stokes operator has a bounded H ^∞-calculus on such domains, and use a general form of Kato’s method. We also obtain information on the corresponding pressure term.     

Singular parabolic equations with measures as initial data

In this paper we study the Cauchy problem for the singular evolution p-Laplacian equations with gradient term and source on the assumption of measures as initial conditions. For the supercritical case q>p−1+p/N, we obtain that for every nonnegative solution there exists a nonnegative Radon measure μ as initial trace and μ has some local regularity.