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Anti-periodic solutions to a parabolic hemivariational inequality

Jong Yeoul Park, Hyun Min Kim, Sun Hye Park (2004)

Kybernetika

In this paper we deal with the anti-periodic boundary value problems with nonlinearity of the form b ( u ) , where b L loc ( R ) . Extending b to be multivalued we obtain the existence of solutions to hemivariational inequality and variational-hemivariational inequality.

Applications of approximate gradient schemes for nonlinear parabolic equations

Robert Eymard, Angela Handlovičová, Raphaèle Herbin, Karol Mikula, Olga Stašová (2015)

Applications of Mathematics

We develop gradient schemes for the approximation of the Perona-Malik equations and nonlinear tensor-diffusion equations. We prove the convergence of these methods to the weak solutions of the corresponding nonlinear PDEs. A particular gradient scheme on rectangular meshes is then studied numerically with respect to experimental order of convergence which shows its second order accuracy. We present also numerical experiments related to image filtering by time-delayed Perona-Malik and tensor diffusion...

Applications of the Carathéodory theorem to PDEs

Konstanty Holly, Joanna Orewczyk (2000)

Annales Polonici Mathematici

We discuss and exploit the Carathéodory theorem on existence and uniqueness of an absolutely continuous solution x: ℐ (⊂ ℝ) → X of a general ODE ( * ) = ( t , x ) for the right-hand side ℱ : dom ℱ ( ⊂ ℝ × X) → X taking values in an arbitrary Banach space X, and a related result concerning an extension of x. We propose a definition of solvability of (*) admitting all connected ℐ and unifying the cases “dom ℱ is open” and “dom ℱ = ℐ × Ω for some Ω ⊂ X”. We show how to use the theorems mentioned above to get approximate...

Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls

Alexander Khapalov (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the one dimensional semilinear reaction-diffusion equation, governed in Ω = (0,1) by controls, supported on any subinterval of (0, 1), which are the functions of time only. Using an asymptotic approach that we have previously introduced in [9], we show that such a system is approximately controllable at any time in both L2(0,1)( and C0[0,1], provided the nonlinear term f = f(x,t, u) grows at infinity no faster than certain power of log |u|. The latter depends on the regularity...

Approximation of the Heaviside function and uniqueness results for a class of quasilinear elliptic-parabolic problems.

G. Gagneux, F. Guerfi (1990)

Revista Matemática de la Universidad Complutense de Madrid

In this paper, we concern ourselves with uniqueness results for an elliptic-parabolic quasilinear partial differential equation describing, for instance, the pressure of a fluid in a three-dimensional porous medium: within the frame of mathematical modeling of the secondary recovery from oil fields, the handling of the component conservation laws leads to a system including such a pressure equation, locally elliptic or parabolic according to the evolution of the gas phase.

Asymptotic analysis for a nonlinear parabolic equation on

Eva Fašangová (1998)

Commentationes Mathematicae Universitatis Carolinae

We show that nonnegative solutions of u t - u x x + f ( u ) = 0 , x , t > 0 , u = α u ¯ , x , t = 0 , supp u ¯ compact either converge to zero, blow up in L 2 -norm, or converge to the ground state when t , where the latter case is a threshold phenomenon when α > 0 varies. The proof is based on the fact that any bounded trajectory converges to a stationary solution. The function f is typically nonlinear but has a sublinear growth at infinity. We also show that for superlinear f it can happen that solutions converge to zero for any α > 0 , provided supp u ¯ is sufficiently small.

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