Search results for "Weak solution"
showing 10 items of 52 documents
Analysis of a parabolic cross-diffusion population model without self-diffusion
2006
Abstract The global existence of non-negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown. The cross-diffusion terms are allowed to be arbitrarily large, whereas the self-diffusion terms are assumed to disappear. The last assumption complicates the analysis since these terms usually provide H 1 estimates of the solutions. The existence proof is based on a positivity-preserving backward Euler–Galerkin approximation, discrete entropy estimates, and L 1 weak compactness arguments. Furthermore, employing the entropy–entropy production method, we show for special stationary solutions that the transient solution converges exponentially fast to its…
On Problems Driven by the (p(·) , q(·)) -Laplace Operator
2020
The aim of this paper is to prove the existence of at least one nontrivial weak solution for equations involving the (p(· ) , q(· ) ) -Laplace operator. The approach is variational and based on the critical point theory.
First and second critical exponents for an inhomogeneous Schrödinger equation with combined nonlinearities
2022
AbstractWe study the large-time behavior of solutions for the inhomogeneous nonlinear Schrödinger equation $$\begin{aligned} iu_t+\Delta u=\lambda |u|^p+\mu |\nabla u|^q+w(x),\quad t>0,\, x\in {\mathbb {R}}^N, \end{aligned}$$ i u t + Δ u = λ | u | p + μ | ∇ u | q + w ( x ) , t > 0 , x ∈ R N , where $$N\ge 1$$ N ≥ 1 , $$p,q>1$$ p , q > 1 , $$\lambda ,\mu \in {\mathbb {C}}$$ λ , μ ∈ C , $$\lambda \ne 0$$ λ ≠ 0 , and $$u(0,\cdot ), w\in L^1_{\mathrm{loc}}({\mathbb {R}}^N,{\mathbb {C}})$$ u ( 0 , · ) , w ∈ L loc 1 ( R N , C ) . We consider both the cases where $$\mu =0$$ μ = 0 and $$\mu \ne 0$$ μ ≠ 0 , respectively. We establish existence/nonexistence of global weak solutions. In ea…
ENTROPY SOLUTIONS IN THE STUDY OF ANTIPLANE SHEAR DEFORMATIONS FOR ELASTIC SOLIDS
2000
The concept of entropy solution was recently introduced in the study of Dirichlet problems for elliptic equations and extended for parabolic equations with nonlinear boundary conditions. The aim of this paper is to use the method of entropy solutions in the study of a new problem which arise in the theory of elasticity. More precisely, we consider here the infinitesimal antiplane shear deformation of a cylindrical elastic body subjected to given forces and in a frictional contact with a rigid foundation. The elastic constitutive law is physically nonlinear and the friction is described by a static law. We present a variational formulation of the model and prove the existence and the uniquen…
De Giorgi–Nash–Moser Theory
2015
We consider the second-order, linear, elliptic equations with divergence structure $$\mathrm{div} (\mathbb{A}(x)\nabla u(x))\;=\;\sum\limits^n_{i,j=1}\;\partial_{x_{i}}(a_{ij}(x)\partial_{x_{j}}u(x))\;=\;0.$$
Stochastic Differential Equations
2020
Stochastic differential equations describe the time evolution of certain continuous n-dimensional Markov processes. In contrast with classical differential equations, in addition to the derivative of the function, there is a term that describes the random fluctuations that are coded as an Ito integral with respect to a Brownian motion. Depending on how seriously we take the concrete Brownian motion as the driving force of the noise, we speak of strong and weak solutions. In the first section, we develop the theory of strong solutions under Lipschitz conditions for the coefficients. In the second section, we develop the so-called (local) martingale problem as a method of establishing weak so…
Stefan-Boltzmann Radiation on Non-convex Surfaces
1997
We consider the stationary heat equation for a non-convex body with Stefan–Boltzmann radiation condition on the surface. The main virtue of the resulting problem is non-locality of the boundary condition. Moreover, the problem is non-linear and in the general case also non-coercive and non-monotone. We show that the boundary value problem has a maximum principle. Hence, we can prove the existence of a weak solution assuming the existence of upper and lower solutions. In the two dimensional case or when a part of the radiation can escape the system we obtain coercivity and stronger existence result. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.
Integro-differential equation modelling heat transfer in conducting, radiating and semitransparent materials
1998
In this work we analyse a model for radiative heat transfer in materials that are conductive, grey and semitransparent. Such materials are for example glass, silicon, water and several gases. The most important feature of the model is the non-local interaction due to exchange of radiation. This, together with non-linearity arising from the well-known Stefan-Boltzmann law, makes the resulting heat equation non-monotone. By analysing the terms related to heat radiation we prove that the operator defining the problem is pseudomonotone. Hence, we can prove the existence of weak solution in the cases where coercivity can be obtained. In the general case, we prove the solvability of the system us…
C1,α-regularity for variational problems in the Heisenberg group
2017
We study the regularity of minima of scalar variational integrals of $p$-growth, $1<p<\infty$, in the Heisenberg group and prove the H\"older continuity of horizontal gradient of minima.
On the regularity of very weak solutions for linear elliptic equations in divergence form
2020
AbstractIn this paper we consider a linear elliptic equation in divergence form $$\begin{aligned} \sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \quad \hbox {in } \Omega . \end{aligned}$$ ∑ i , j D j ( a ij ( x ) D i u ) = 0 in Ω . Assuming the coefficients $$a_{ij}$$ a ij in $$W^{1,n}(\Omega )$$ W 1 , n ( Ω ) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution $$u\in L^{n'}_\mathrm{loc}(\Omega )$$ u ∈ L loc n ′ ( Ω ) of (0.1) is actually a weak solution in $$W^{1,2}_\mathrm{loc}(\Omega )$$ W loc 1 , 2 ( Ω ) .