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In this paper we study the stable set of the gradient flow associated with a critical point of an analytic function. In particular we present simple topological conditions which imply that this set contains an infinite family of trajectories, or has a non-empty interior., arXiv admin note: substantial text overlap with arXiv:1912.09149

We consider semilinear Neumann problems at resonance. We are concerned with two distinct cases. In the first one, the potential function is unbounded and indefinite. In the second case, the potential function is bounded and the reaction exhibits a concave parametric term near the origin. We prove two existence theorems producing sign-changing solutions. The first main result is a complement of the results obtained by Papageorgiou and Radulescu (2015) [22]. The second theorem gives an answer to the open question raised by Candito et al. (2016) [4]. Our approach uses variational methods together with flow invariance arguments.

We consider a parametric Neumann problem with nonhomogeneous differential operator and critical growth. Combining variational methods based on critical point theory, with suitable truncation techniques and flow invariance arguments, we show that for all large λ , the problem has at least three nontrivial smooth solutions, two of constant sign (one positive, the other negative) and the third nodal. We also study the asymptotic behavior of all solutions obtained when λ converges to infinity. The interesting point is that we do not impose any restrictions to the behavior of the nonlinear term f at infinity. Our work unifies and sharply improves several recent papers.

We prove the existence of global in time weak nonnegative solutions to a family of nonlinear fourth-order evolution equations, parametrized by a real parameter q∈(0, 1], which includes the well known thin-film (q=1/2) and the Derrida–Lebowitz– Speer–Spohn (DLSS) equation (q=1), subject to periodic boundary conditions in one spatial dimension. In contrast to the gradient flow approach in [25], our method relies on dissipation property of the corresponding entropy functionals (Tsallis entropies) resulting in required a priori estimates, and extends the existence result from [25] to a wider range of the family members, namely to 0

We establish an algebraic contraction rate in a modified Wasserstein distance for solutions of scalar conservation laws with uniformly convex flux. We also show that our estimate is optimal w.r.t. scaling in time and discuss why it gives non-trivial information in relation to the stability of the rarefaction wave.

In this paper, we prove that the Jordan–Kinderlehrer–Otto scheme for a family of linear parabolic equations on the flat torus converges uniformly in space.

In this paper, we consider the existence of positive, negative and sign-changing solutions for some fourth order semilinear elliptic boundary value problems. We present new results on invariant sets of the gradient flows of the corresponding variational functionals. The structure of the invariant sets will be built into minimax procedures to construct the sign-changing solutions.

The purpose of this paper is twofold. The first is to remove a possible ill-posedness related to a local minimax method developed in SIAM J. Sci. Comput. 23 (2001) 840–865, SIAM J. Sci. Comput. 24 (2002) 840–865 and the second is to provide a local characterization for nonminimax type saddle points. To do so, a local L -⊥ selection is defined and a necessary and sufficient condition for a saddle point is established, which leads to a min-orthogonal method. Those results exceed the scope of a minimax principle, the most popular approach in critical point theory. An example is given to illustrate the new theory. With this local characterization, the local minimax method in SIAM J. Sci. Comput. 23 (2001) 840–865, SIAM J. Sci. Comput. 24 (2002) 840–865 is generalized to a local min-orthogonal method for finding multiple saddle points. In a subsequent paper, this approach is applied to define a modified pseudo gradient (flow) of a functional for finding multiple saddle points in Banach spaces.

The shock wave in a viscous gas which is treated as a strong discontinuity is unstable against small perturbations [A.M. Blokhin, On stability of shock waves in a compressible viscous gas, Matematiche LVII (I) (2002) 3–19]. We suggest such additional boundary conditions that a modified (with account to these conditions) linear initial-boundary value problem on stability of the shock wave does not admit Hadamard-type ill-posedness examples.