Combined effects for a class of fractional variational inequalities

In this paper, we revisit the following nonlocal Kirchhoff diffusion problem:∂tu+M([u]s2)LKu=|u|p−2u,inΩ×R+,u(x,t)=0,in(RN\\Ω)×R+,u(x,0)=u0(x),inΩ,where is a bounded domain with Lipschitz boundary, [u]s is the Gagliardo seminorm of u, 0 < s < min{1, N/2}, is a nonlocal integro-differential operator defined in (), which generalizes the fractional Laplace operator (−Δ)s, u0 : Ω → [0, +∞) is the initial function, M : [0, +∞) → [0, +∞) is a continuous function and there exist two constants θ > 1 and m0 > 0 such thatM(σ)⩾m0σθ−1,∀σ∈[0,+∞).This problem has been investigated by Xiang, Rădulescu and Zhang in [], and Ding and Zhou in [] by using potential well method. If , in [], the authors showed the existence of a nontrivial, nonnegative global weak solution, where . However, if , these two papers only studied the model in special cases, the details are as follows: in [], the blow-up conditions for nontrivial, nonnegative weak solution were obtained when J(u0) < 0; in [], the global existence and blow-up conditions for nontrivial, nonnegative weak solution were obtained when J(u0) ⩽ d and M(σ) = m0σθ−1, where J(u0) denotes the initial energy and d > 0 denotes the depth of the potential well (see ()). The main purpose of this paper is to extend the above results to the general case M(σ) ⩾ m0σθ−1, , and the conditions on global existence and finite time blow-up are obtained. Furthermore, the decay estimates for global weak solutions, the growth estimates for blow-up solutions, the upper and lower bounds of blow-up time to blow-up solutions, the behavior of the energy functional as t → T (where T denotes the blow-up time) are studied. Moreover, some blow-up conditions independent of d and some equivalent conditions for the weak solutions existing globally or blowing up in finite time are investigated. Finally, the global existence and finite time blow-up results with high initial energy (i.e., J(u0) > d) are obtained.

In this paper, we study the following diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator where [u]s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz boundary, , is a nonlocal integro-differential operator defined in (), which generalizes the fractional Laplace operator , is the initial function, and is a continuous function and there exist two constants m0 > 0 and such that As is well-known, the nonlocal Kirchhoff problem was first introduced and motivated in Fiscella and Valdinoci (2014 Nonlinear Anal. 94 156–70) and the above problem was studied by Xiang et al (2018 Nonlinearity 31 3228–50), the main results of Xiang et al (2018 Nonlinearity 31 3228–50) are as follows: The local existence of nontrivial, nonnegative weak solution for , where . The blow-up conditions for nontrivial, nonnegative weak solution when J(u0) < 0, where J(u0) denotes the initial energy.The main purpose of this paper is to extend the above results and we get: The global existence of nontrivial, nonnegative weak solution for any . The global existence and blow-up conditions for nontrivial, nonnegative weak solution when for the case , where d is a positive constant given in ().

Semilinear elliptic variational inequalities with dependence on the gradient via Mountain Pass techniques

We consider the effect of a second-order 'porous media' term on the evolution of weak solutions of the fourth-order degenerate diffusion equation ht=- Del .(hn Del Delta h- Del hm) in one space dimension. The equation without the second-order term is derived from a 'lubrication approximation' and models surface tension dominated motion of thin viscous films and spreading droplets. Here h (x,t) is the thickness of the film, and the physical problem corresponds to n=3. For simplicity, we consider periodic boundary conditions which has the physical interpretation of modelling a periodic array of droplets. In a previous work we studied the above equation without the second-order 'porous media' term. In particular we showed the existence of non-negative weak solutions with increasing support for 0<n<3 but the techniques failed for n>or=3. This is consistent with the fact that, in this case, non-negative self-similar source-type solutions do not exist for n>or=3. In this work, we discuss a physical justification for the 'porous media' term when n=3 and 1<m<2. We propose such behaviour as a cut off of the singular 'disjoining pressure' modelling long range van der Waals interactions. For all n>0 and 1<m<2, we discuss possible behaviour at the edge of the support of the solution via leading order asymptotic analysis of travelling wave solutions. This analysis predicts a certain 'competition' between the second- and fourth-order terms. We present rigorous weak existence theory for the above equation for all n>0 and 1<m<2. In particular, the presence of a second-order 'porous media' term in the above equation yields non-negative weak solutions that converge to their mean as t to infinity and that have additional regularity. Moreover, we show that there exists a time T* after which the weak solution is a positive strong solution. For n>3/2, we show that the regularity of the weak solutions is in exact agreement with that predicted by the asymptotics. Finally, we present several numerical computations of solutions. The simulations use a weighted implicit-explicit scheme on a dynamically adaptive mesh. The numerics suggest that the weak solution described by our existence theory has compact support with a finite speed of propagation. The data confirms the local 'power law' behaviour at the edge of the support predicted by asymptotics.

Weak Sharp Solutions of Variational Inequalities

This work aims is to study a nonlinear second-order boundary value differential elliptic problem in one dimension where the nonlinearity concerns the solution and its first derivative. We assume that the source term can be non-smooth and the nonlinearity can grow faster than quadratic. First, we show the existence of a non-negative weak solution if we assume the existence of a super-solution. Second, we present a numerical algorithm to compute an approximation of the non-negative weak solution. The proposed algorithm is decomposed in two steps, the first one is devoted to computing a super-solution, and in the second one, the algorithm computes a sequence of solutions of an intermediate problem obtained by using the Yosida approximation of the nonlinearity. This sequence converges to the non-negative weak solution of the nonlinear equation. The numerical method is an application of the Newton method to the discretized version of the problem, but at each iteration, the resulting system can be indefinite. To overcome this difficulty, we introduce an adaptive non-overlapping domain decomposition method.

In this paper, we study the existence of a weak solution for a class of elliptic systems involving nonhomogeneous operators. By using the Mountain-Pass theorem and establishing a compactness result, we prove the existence of nontrivial and non-negative weak solution. Our results generalize previous work by considering broader classes of operators and nonlinearities that satisfy superlinear growth conditions.

Non-negative global weak solutions for a degenerated parabolic system approximating the two-phase Stokes problem

We prove the global existence of non-negative weak solutions for a strongly coupled, fourth-order degenerate parabolic system governing the motion of two thin fluid layers in a porous medium when capillarity is the sole driving mechanism.

We study the homogeneous Dirichlet problem for a class of nonlinear anisotropic parabolic double phase equations with nonstandard growth conditions. We prove the existence of a unique nonnegative weak solution in suitable Orlicz-Sobolev spaces, and derive the global boundedness of the weak solution.

The subject of this thesis is mathematical modeling and analysis of dewetting dynamics and equilibrium patterns of two-layer thin liquid films. We derive systems of coupled thin-film equations for two immiscible liquid layers on a solid substrate that take interfacial slip and intermolecular forces into account. By using these thin-film models, we study the stability of two-layer systems and investigate its dependence on the order of magnitude of slip. The resulting dispersion relations exhibit two local maxima. One appears for small wavenumbers and mainly affects the liquid-gas interface, the other arises for moderate wavenumbers and its associated perturbations have higher amplitudes at the liquid-liquid interface. Varying the slip lengths at the interfaces influences the two maxima, in particular, it may lead to a transition of the dominant wavelength and thereby change the spinodal patterns significantly. Then, we investigate stationary states of flows of thin liquid two-layers via the derived models. Assuming a negative spreading coefficient, which emerges from the intermolecular potential, the energy to the system favors the lower liquid layer to be only partially covered by the upper liquid. On the other hand, the intermolecur forces lead to an ultra-thin layer of thickness h∗. For the stationary problem to the thin-film models we prove existence of solutions. Moreover, in the limit h∗ to 0 we apply matched asymptotic analysis to derive sharp-interface models and corresponding contact angles, i.e. the Neumann triangle. We use Γ-convergence to derive a sharp-interface energy rigorously in this limit and determine the minimizers to this energy. These minimizers agree with the solutions obtained by matched asymptotics. Furthermore, we present comparisons of numerical simulations with experimental results. We also show existence of non-negative global weak solutions to the derived thin-film models for small and moderate slip lengths in the case that intermolecular forces are neglected. In addition, we prove existence of positive smooth solutions when intermolecular forces between the liquids and between the lower liquid and the solid substrate are taken into account. For the thin-film model which allows for large slip we show that weak solutions are stable under perturbations of the initial data.

A basic theory for initial value problems of first order ordinary differential equations with Lp-Carathéodory functions and applications

&lt;abstract&gt;&lt;p&gt;Compared to the standard variational inequalities, inverse variational inequalities are more suitable for pricing American options with indefinite payoff. This paper investigated the initial-boundary value problem of inverse variational inequalities constituted by a class of non-divergence type parabolic operators. We established the existence and Hölder continuity of weak solutions. Since the comparison principle in the case of standard variational inequalities is no longer applicable, we constructed an integral inequality using differential inequalities to determine the global upper bound of the solution. By combining it with the continuous method, we obtained the existence of weak solutions. Additionally, by employing truncation factors, we obtained the lower bound of weak solutions in the cylindrical subdomain, thereby obtaining the Hölder continuity.&lt;/p&gt;&lt;/abstract&gt;

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $C^{1,1}$ boundary. We consider problems of the form $-\Delta u=\chi_{\left\{ u&gt;0\right\}}\left( au^{-\alpha}-g\left( .,u\right) \right) $ in $\Omega,$ $u=0$ on $\partial\Omega,$ $u\geq0$ in $\Omega,$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $0\not \equiv a\in L^{\infty}\left( \Omega\right) ,$ $\alpha\in\left( 0,1\right) ,$ and $g:\Omega\times\left[ 0,\infty\right) \rightarrow\mathbb{R}$ is a nonnegative Carathéodory function. We prove, under suitable assumptions on $a$ and $g,$ the existence of nontrivial and nonnegative weak solutions $u\in H_{0}^{1}\left( \Omega\right) \cap L^{\infty}\left( \Omega\right) $ of the stated problem. Under additional assumptions, the positivity, $a.e.$ in $\Omega,$ of the found solution $u$, is also proved.

Nonlocal Harnack inequalities for nonlocal heat equations