Nonlinear Differential Equations and Applications NoDEA

We consider a strongly damped semilinear equation arising in the theory of elastic beams. We prove the existence of an exponentially attracting finite-dimensional vector space in the Hilbert space of the solutions (a so-called inertial manifold), and provide some numerical estimates of the dimension of the manifold.

In this work we extend a recent result by Dyda et al. (J Lond Math Soc 95(2):500–518, 2017) to dimension 3.

We consider some questions on G-convergence of a sequence of Nemytskii maximal monotone operators defined on the space of square integrable functions acting from a real interval to a separable Hilbert space. Every Nemytskii operator is generated by a time dependent family of maximal monotone operators. The values of these maximal monotone operators are normal cones of closed convex sets. These sets are values of a multivalued mapping from a real interval to a separable Hilbert space. The convergence of Nemytskii maximal monotone operators is used to study the dependence of solutions to sweeping processes on a parameter.

In this short note, we consider the elliptic problem $$\begin{aligned} \lambda \phi + \Delta \phi = \eta |\phi |^\sigma \phi ,\quad \phi \big |_{\partial \Omega }=0,\quad \lambda , \eta \in {{\mathbb {C}}}, \end{aligned}$$ on a smooth domain $$\Omega \subset {{\mathbb {R}}}^N$$ , $$N\geqslant 1$$ . The presence of complex coefficients, motivated by the study of complex Ginzburg-Landau equations, breaks down the variational structure of the equation. We study the existence of nontrivial solutions as bifurcations from the trivial solution. More precisely, we characterize the bifurcation branches starting from eigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. This allows us to discuss the nature of such bifurcations in some specific cases. We conclude with the stability analysis of these branches under the complex Ginzburg-Landau flow.

The curve shortening problem for a graph under Robin boundary condition is studied in this paper. The large time behavior of the global solution is shown to depend critically on the parameters in the boundary condition. Some asymptotic behavior of the solution is also discussed.

Isothermal compressible two-phase flows in a capillary are modeled with and without phase transition in the presence of gravity, employing Darcy’s law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria with flat interface are identified. It is shown that the problems are well-posed in an $$L_p$$ -setting and generate local semiflows in the proper state manifolds. The main result concerns the stability of equilibria with flat interface, i.e. the Rayleigh–Taylor instability.

We investigate rates of convergence for two approximation schemes of time-independent and time-dependent Hamilton–Jacobi equations with Kirchoff junction conditions. We analyze the vanishing viscosity limit and monotone finite-difference schemes. Following recent work of Lions and Souganidis, we impose no convexity assumptions on the Hamiltonians. For stationary Hamilton–Jacobi equations, we obtain the classical $$\epsilon ^{\frac{1}{2}}$$ rate, while we obtain an $$\epsilon ^{\frac{1}{7}}$$ rate for approximations of the Cauchy problem. In addition, we present a number of new techniques of independent interest, including a quantified comparison proof for the Cauchy problem and an equivalent definition of the Kirchoff junction condition.

Điểm mới chính của bài viết này là tiết lộ một dạng yếu của chuyển động cơ thể rắn sinh ra từ sự tiến triển của các ranh giới tự do trong các dòng chảy visco-plastic được điều khiển bởi toán tử Norton–Hoff không nén với các hệ số phi trụ. Chúng tôi cung cấp kết quả tồn tại của một giao diện giữa hai chất lỏng không thể trộn lẫn bằng cách sử dụng lý thuyết tiến triển không trơn tru. Chúng tôi chứng minh rằng dòng chảy của chất lỏng được chuyển đổi thành một cơ thể rắn khi độ nhớt của nó đủ lớn. Các kết quả đã thiết lập là các biến thể hoặc mở rộng của các mô hình hiện có.

In this paper, we study an optimal control problem associated to the conformal CR sub-Laplacian obstacle problem on a compact pseudohermitian manifold. When the CR Yamabe constant is positive, we show that the optimal controls are equal to their associated optimal states and show the existence of a smooth optimal control which induces a conformal contact form with constant Webster scalar curvature.