Journal of Earthquake Engineering

Abstract We prove existence and comparison results for multi-valued variational inequalities in a bounded domain $$\Omega $$ Ω of the form $$\begin{aligned} u\in K{:}\, 0 \in Au+\partial I_K(u)+{\mathcal {F}}(u)+{\mathcal {F}}_\Gamma (u)\quad \text {in }W^{1, {\mathcal {H}}}(\Omega )^*, \end{aligned}$$ u ∈ K : 0 ∈ A u + ∂ I K ( u ) + F ( u ) + F Γ ( u ) in W 1 , H ( Ω ) ∗ , where $$A{:}\,W^{1, {\mathcal {H}}}(\Omega ) \rightarrow W^{1, {\mathcal {H}}}(\Omega )^*$$ A : W 1 , H ( Ω ) → W 1 , H ( Ω ) ∗ given by $$\begin{aligned} Au:=-\text {div}\left( |\nabla u|^{p(x)-2} \nabla u+ \mu (x) |\nabla u|^{q(x)-2} \nabla u\right) \end{aligned}$$ A u : = - div | ∇ u | p ( x ) - 2 ∇ u + μ ( x ) | ∇ u | q ( x ) - 2 ∇ u for $$u \in W^{1, {\mathcal {H}}}(\Omega )$$ u ∈ W 1 , H ( Ω ) , is the double phase operator with variable exponents and $$W^{1, {\mathcal {H}}}(\Omega )$$ W 1 , H ( Ω ) is the associated Musielak–Orlicz Sobolev space. First, an existence result is proved under some weak coercivity condition. Our main focus aims at the treatment of the problem under consideration when coercivity fails. To this end we establish the method of sub–super-solution for the multi-valued variational inequality in the space $$W^{1, {\mathcal {H}}}(\Omega )$$ W 1 , H ( Ω ) based on appropriately defined sub- and super-solutions, which yields the existence of solutions within an ordered interval of sub–super-solution. Moreover, the existence of extremal solutions will be shown provided the closed, convex subset K of $$W^{1, {\mathcal {H}}}(\Omega )$$ W 1 , H ( Ω ) satisfies a lattice condition. As an application of the sub–super-solution method we are able to show that a class of generalized variational–hemivariational inequalities with a leading double phase operator are included as a special case of the multi-valued variational inequality considered here. Based on a fixed point argument, we also study the case when the corresponding Nemytskij operators $${\mathcal {F}}, {\mathcal {F}}_\Gamma $$ F , F Γ need not be continuous. At the end, we give an example of the construction of sub- and supersolutions related to the problem above.

In this research, we investigate a general shape optimization problem in which the state equation is expressed using a nonlocal and nonlinear operator. We prove the existence of a minimum point for a functional F defined on the family of all ’quasi-open’ subsets of a bounded open set $$\Omega $$ in $$\mathbb {R}^n$$ . This is ensured under the condition that F demonstrates decreasing behavior concerning set inclusion and is lower semicontinuous with respect to a suitable topology associated with the fractional p-Laplacian under Dirichlet boundary conditions. Moreover, we study the asymptotic behavior of the solutions when $$s\rightarrow 1$$ and extend this result to the anisotropic case.

Abstract In parabolic or hyperbolic PDEs, solutions which remain uniformly bounded for all real times $$t=r\in \mathbb {R}$$ t = r ∈ R are often called PDE entire or eternal. For a nonlinear example, consider the quadratic parabolic PDE $$\begin{aligned} w_t=w_{xx}+6w^2-\lambda , \end{aligned}$$ w t = w xx + 6 w 2 - λ , for $$0<x<\tfrac{1}{2}$$ 0 < x < 1 2 , under Neumann boundary conditions. By its gradient-like structure, all real eternal non-equilibrium orbits $$\Gamma (r)$$ Γ ( r ) of (*) are heteroclinic among equilibria $$w=W_n(x)$$ w = W n ( x ) . For parameters $$\lambda >0$$ λ > 0 , the trivial homogeneous equilibria are locally asymptotically stable $$W_0=-\sqrt{\lambda /6}$$ W 0 = - λ / 6 , and $$W_\infty =+\sqrt{\lambda /6}$$ W ∞ = + λ / 6 of unstable dimension (Morse index) $$i(W_\infty )=1,2,3,\ldots $$ i ( W ∞ ) = 1 , 2 , 3 , … , depending on $$\lambda $$ λ . All nontrivial real $$W_n$$ W n are rescaled and properly translated real-valued Weierstrass elliptic functions with Morse index $$i(W_n)=n$$ i ( W n ) = n . We show that the complex time extensions $$\Gamma (r+\textrm{i}s)$$ Γ ( r + i s ) , of analytic real heteroclinic orbits $$\Gamma (r)$$ Γ ( r ) towards $$W_0$$ W 0 , are not complex entire. For example, consider the time-reversible complex-valued solution $$\psi (s)=\Gamma (r_0-\textrm{i}s)$$ ψ ( s ) = Γ ( r 0 - i s ) of the nonlinear and nonconservative quadratic Schrödinger equation $$\begin{aligned} \textrm{i}\psi _s=\psi _{xx}+6\psi ^2-\lambda \end{aligned}$$ i ψ s = ψ xx + 6 ψ 2 - λ with real initial condition $$\psi _0=\Gamma (r_0)$$ ψ 0 = Γ ( r 0 ) . Then there exist real $$r_0$$ r 0 such that $$\psi (s)$$ ψ ( s ) blows up at some finite real times $$\pm s^*\ne 0$$ ± s ∗ ≠ 0 . Abstractly, our results are formulated in the setting of analytic semigroups. They are based on Poincaré non-resonance of unstable eigenvalues at equilibria $$W_n$$ W n , near pitchfork bifurcation. Technically, we have to except discrete sets of parameters $$\lambda $$ λ , and are currently limited to unstable dimensions $$i(W_n)\le 22$$ i ( W n ) ≤ 22 , or to fast unstable manifolds of dimensions $$d<1+\tfrac{1}{\sqrt{2}}i(W_n)$$ d < 1 + 1 2 i ( W n ) .

For $$p>2$$ , the equation $$\begin{aligned} u_t = u^p u_{xx}, \qquad x\in \mathbb {R}, \ t\in \mathbb {R}, \end{aligned}$$ is shown to admit positive and spatially increasing smooth solutions on all of $$\mathbb {R}\times \mathbb {R}$$ which are precisely of the form of an accelerating wave for $$t<0$$ , and of a wave slowing down for $$t>0$$ . These solutions satisfy $$u(\cdot ,t)\rightarrow 0$$ in $$L^\infty _{loc}(\mathbb {R})$$ as $$t\rightarrow + \infty $$ and as $$t\rightarrow -\infty $$ , and exhibit a yet apparently undiscovered phenomenon of transient rapid spatial growth, in the sense that $$\begin{aligned} \lim _{x\rightarrow +\infty } x^{-1} u(x,t) \quad \text{ exists } \text{ for } \text{ all } t<0, \end{aligned}$$ that $$\begin{aligned} \lim _{x\rightarrow +\infty } x^{-\frac{2}{p}} u(x,t) \quad \text{ exists } \text{ for } \text{ all } t>0, \end{aligned}$$ but that $$\begin{aligned} u(x,0)=K e^{\alpha x} \qquad \text{ for } \text{ all } x\in \mathbb {R}\end{aligned}$$ with some $$K>0$$ and $$\alpha >0$$ .

In this paper, we use a Kajikiya’s version of the symmetric mountain pass lemma and Moser iteration method to prove the existence of infinitely many small solutions for a class of nonlinear fractional Kirchhoff–Schrödinger-Poisson equations under some new and weak sublinear conditions on the nonlinear term. Some examples are given to illustrate our main theoretical results.

The main goal of this paper is to study the asymptotic behavior of a weakly damped forced fractional nonlinear Klein–Gordon–Schrödinger system in one dimensional unbounded domain. We prove the existence of a global attractor $$\mathcal {A}_{\alpha }\; \left( \alpha \in (\frac{1}{2},1)\right) $$ of the systems of the fractional nonlinear Klein–Gordon–Schrödinger (FNLKGS) equations in $$H^{\alpha }(\mathbb {R})\times H^{\alpha }(\mathbb {R})\times L^2(\mathbb {R})$$ and more particularly that this attractor is in fact a compact set of $$H^{2\alpha }(\mathbb {R})\times H^{2\alpha }(\mathbb {R})\times H^{\alpha }(\mathbb {R})$$ .

In this paper, we study a class of parametric anisotropic elliptic equations with variable exponents where the nonlinearity may depend on the gradient of the solution. We prove the existence of the solution using the surjectivity result of pseudomonotone operators, and under additional conditions on the data, we show that the solution is unique. Moreover, we establish the existence of at least three weak solutions using the direct Ricceri variational principle when the nonlinearity does not depend on the gradient.

We investigate the diffusive Hamilton–Jacobi equation $$\begin{aligned} u_t-\Delta u = |\nabla u|^p \end{aligned}$$ with $$p>1$$ , in a smooth bounded domain of $${\mathbb {R}^n}$$ with homogeneous Neumann boundary conditions and $$W^{1,\infty }$$ initial data. We show that all solutions exist globally, are bounded and converge in $$W^{1,\infty }$$ norm to a constant as $$t\rightarrow \infty $$ , with a uniform exponential rate of convergence given by the second Neumann eigenvalue. This improves previously known results, which provided only an upper polynomial bound on the rate of convergence and required the convexity of the domain. Furthermore, we extend these results to a rather large class of nonlinearities $$F(\nabla u)$$ instead of  $$|\nabla u|^p$$ .

In this paper, we consider a nonlinear mathematical model which describes the deformation of a viscous body in a three-dimensional thin domain with the presence of Tresca’s friction law. We examine the asymptotic behavior of the weak solutions when one dimension of the domain tends to zero. The two-dimensional limit problem has been justified. The uniqueness result of the displacement and the limit form of the Tresca boundary conditions are obtained.

In this paper, we study a parabolic free boundary problem in an exterior domain $$\begin{aligned} {\left\{ \begin{array}{ll} F(D^2u)-\partial _tu=u^a\chi _{\{u>0\}}& \text {in }({{\mathbb {R}}}^n\setminus K)\times (0,\infty ),\\ u=u_0& \text {on }\{t=0\},\\ |\nabla u|=u=0& \text {on }\partial \Omega \cap ({{\mathbb {R}}}^n\times (0,\infty )),\\ u=1& \text {in }K\times [0,\infty ). \end{array}\right. } \end{aligned}$$ Here, a belongs to the interval $$(-1,0)$$ , K is a (given) convex compact set in $${{\mathbb {R}}}^n$$ , $$\Omega =\{u>0\}\supset K\times (0,\infty )$$ is an unknown set, and F denotes a fully nonlinear operator. Assuming a suitable condition on the initial value $$u_0$$ , we prove the existence of a nonnegative quasiconcave solution to the aforementioned problem, which exhibits monotone non-decreasing behavior over time.

In this paper, we establish the existence of weak solutions for a parabolic problem involving the fractional q(z)-Laplacian operator $$(-\Delta )_{q(z)}^\sigma $$ . The main results are derived using the Young measures method combined with generalized Lebesgue and Sobolev spaces with variable exponents, alongside the Galerkin approximation. Our results expand upon and broaden several recent studies in this area of literature

The paper deals with a parabolic boundary value problem that describes the chaotic dynamics of a single polymer chain in a liquid. The time in the parabolic equation plays the role of the arc length parameter along the chain and corresponds to the link number. The equation includes a so called interaction (between the links) potential that has a double non-locality. It depends on the integrals of the solution over the entire time interval and over the entire space domain where the problem is being solved. Moreover, the space non-locality is in a denominator, which causes additional difficulties related to the possible vanishing of the integral. It is managed to prove that this integral does not vanish, if the interaction potential is bounded on an open part of the domain. This part can be very small, but not empty. This condition implies that the potential is a function of two variables. The first one is the space variable and the second depends on the solution. With respect to the second variable, the potential satisfies fairly general conditions and can be a bounded from below continuous function with an arbitrary growth at infinity, which entails that the interaction term is not a lower order term in the equation. The existence of a weak solution of the initial boundary value problem is proven.