Nonlinear Differential Equations and Applications NoDEA

In this paper we introduce an isospectral flow (Lax flow) that deforms real Hessenberg matrices to Jacobi matrices isospectrally. The Lax flow is given by $$\frac{dA}{dt} = [[A^T, A]_{du}, A],$$ where brackets indicate the usual matrix commutator, [A, B] : = AB−BA, A T is the transpose of A and the matrix [A T , A] du is the matrix equal to [A T , A] along diagonal and upper triangular entries and zero below diagonal. We prove that if the initial condition A 0 is upper Hessenberg with simple spectrum and subdiagonal elements different from zero, then $${\lim_{t\rightarrow +\infty}A(t)}$$ exists, it is a tridiagonal symmetric matrix isospectral to A 0 and it has the same sign pattern in the codiagonal elements as the initial condition A 0. Moreover we prove that the rate of convergence is exponential and that this system is the solution of an infinite horizon optimal control problem. Some simulations are provided to highlight some aspects of this nonlinear system and to provide possible extensions to its applicability.

In this paper we obtain the local Hölder regularity of the gradient of weak solutions for the general parabolic p(x, t)-Laplacian equations $$u_t-\text{div}\ \mathcal{A}\left( \nabla u, x, t \right)\,=\,\text{div} \left(|\mathrm{\bf f}|^{p(x, t)-2} \mathrm{\bf f}\right),$$ provided p(x, t), $${\mathcal{A}}$$ and $${\mathrm{\bf f}}$$ satisfy some proper conditions. More precisely, we shall prove that $$\nabla u \in C_{loc}^{0;\alpha,\alpha/2}(\Omega_T)\,\mbox{for some} \, \, \alpha \in (0, 1). $$

We derive conditions for the positivity and boundedness of the Green functions of the higher order linear nonautonomous ODE. By virtue of these conditions, the existence of positive solutions for a class of nonlinear equations is proved. In addition, upper and lower estimates for the Green functions are established. Moreover, it is shown that nonlinear equations, having separated nonautonomous linear parts, satisfy the generalized Aizerman hypothesis on absolute stability, if they have the positive Green functions.

In this work we prove the existence and uniqueness of the strong solution of the shell model of turbulence perturbed by Lévy noise. The local monotonicity arguments have been exploited in the proofs.

We prove several existence results for eigenvalue problems involving the p-Laplacian and a nonlinear boundary condition on unbounded domains. We treat the non-degenerate subcritical case and the solutions are found in an appropriate weighted Sobolev space.

In this work we investigate the existence and multiplicity of solutions for a class of nonlocal problems involving the fractional N/s-Laplacian and nonlinearities that may have exponential growth of Trudinger–Moser type. First, by using the constraint variational method, the quantitative deformation lemma and the fractional Trudinger–Moser inequality, we establish the existence of a least energy nodal solution. Next, minimax techniques are exploited to prove the existence of one nonnegative and of one nonpositive ground state solution. In a third stage, we show that the energy of the nodal solution is strictly larger than twice the ground state energy.

Bài báo này trình bày một phân loại đầy đủ về hành vi ngắn hạn của các bề mặt giao diện trong bài toán Cauchy cho phương trình vi phân phân cấp phi tuyến bậc hai suy thoái $$\begin{aligned} u_t-\Delta u^m +b u^\beta =0, \quad x\in {\mathbb {R}}^N, 01, C,\alpha , \beta >0, b \in {\mathbb {R}}$$. Bề mặt giao diện $$t=\eta (x)$$ có thể co lại, mở rộng hoặc giữ nguyên tùy thuộc vào sức mạnh tương đối của các thuật ngữ khuếch tán và phản ứng gần biên hỗ trợ, được biểu diễn bằng các tham số $$m,\beta , \alpha , sign\ b$$ và C. Trong tất cả các trường hợp, chúng tôi chứng minh công thức rõ ràng cho sự tiệm cận của giao diện và giải pháp cục bộ gần giao diện.