Nonlinear Analysis-Theory Methods & Applications最新文献

We prove existence for a class of signed Radon measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz continuous. Existence is proven by a constructive procedure which makes use of a suitable family of approximating problems. Relevant qualitative properties of such constructed solutions are pointed out.

The aim of this paper is to investigate the minimization problem related to a Ginzburg–Landau energy functional, where in particular a nonlinear diffusion of mean curvature-type is considered, together with a classical double well potential. A careful analysis of the corresponding Euler–Lagrange equation, equipped with natural boundary conditions and mass constraint, leads to the existence of an unique Maxwell solution, namely a monotone increasing solution obtained for small diffusion and close to the so-called Maxwell point. Then, it is shown that this particular solution (and its reversal) has least energy among all the stationary points satisfying the given mass constraint. Moreover, as the viscosity parameter tends to zero, it converges to the increasing (decreasing for the reversal) single interface solution, namely the constrained minimizer of the corresponding energy without diffusion. Connections with Cahn–Hilliard models, obtained in terms of variational derivatives of the total free energy considered here, are also presented.

We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish that the global entropy weak solutions, constructed in Amadori and Christoforou (2022) to the Cauchy problem for any $BV$ initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein, admit unconditional time-asymptotic flocking without any further assumptions on the initial data. In addition, we show that the convergence to a flocking profile occurs exponentially fast.

We study the exact effect of the anisotropic convexity of domains on the boundary estimate for two Monge–Ampère Equations: one is singular which is from the proper affine hyperspheres with constant mean curvature; the other is degenerate which is from the Monge–Ampère eigenvalue problem. As a result, we obtain the sharp boundary estimates and the optimal global Hölder regularity for the two equations.

We prove Ambrosetti–Prodi type results for periodic solutions of some one-dimensional nonlinear problems that can have drift term whose principal operator is the fractional Laplacian of order $s\in (0,1)$. We establish conditions for the existence and nonexistence of solutions of those problems. The proofs of the existence results are based on the sub-supersolution method combined with topological degree type arguments. We also obtain a priori bounds in order to get multiplicity results. We also prove that the solutions are $C^{1,\alpha }$ under some regularity assumptions in the nonlinearities, that is, the solutions of the mentioned equations are classical. We finish the work obtaining Ambrosetti-Prodi-type results for a problem with singular nonlinearities.

We give another proof of a theorem of Rabinowitz and Stredulinsky obtaining infinite transition solutions for an Allen–Cahn equation. Rabinowitz and Stredulinsky have constructed infinite transition solutions as locally minimal solutions, but it is still an interesting question to establish these solutions by other method. Our result may attract the interest of constructing solutions with the shape of locally minimal solutions of Rabinowitz and Stredulinsky for problems defined on descrete group.

In this work we study the existence and uniqueness of a positive, as well as a sign-changing steady-state solution of the degenerate logistic equation with a non-homogeneous superlinear term. Our outcome on a solution that changes sign, defined in higher dimensions, contribute to the existing literature of a few results for the problem, mostly developed in one dimension. We apply variational techniques, in particular the problem constrained to the Nehari manifold, and investigate how it changes as the parameter $\lambda$ in the equation or the function $b$ vary, affecting the existence and non-existence of solutions of the elliptic problem.

The existence of a solution to a quasilinear system of degenerate equations is proved, assuming that the datum has an intermediate degree of summability and that the off-diagonal coefficients have a support contained in a crossed staircase set. The support required in this paper is larger than the one assumed in a previous work.

Following the previous part of our study on unsteady non-Newtonian fluid flows with boundary conditions of friction type we consider in this paper the case of pseudo-plastic (shear thinning) fluids. The problem is described by a $p$-Laplacian non-stationary Stokes system with $p<2$ and we assume that the fluid is subjected to mixed boundary conditions, namely non-homogeneous Dirichlet boundary conditions on a part of the boundary and a slip fluid-solid interface law of friction type on another part of the boundary. Hence the fluid velocity should belong to a subspace of $L^p\left(0,T;\left(W^{1,p}{\left(\Omega \right)}^3\right)\right)$, where $\Omega$ is the flow domain and $T>0$, and satisfy a non-linear parabolic variational inequality. In order to solve this problem we introduce first a vanishing viscosity technique which allows us to consider an auxiliary problem formulated in $L^{p^{\prime }}\left(0,T;\left(W^{1,p^{\prime }}{\left(\Omega \right)}^3\right)\right)$ with $p^{\prime }>2$ the conjugate number of $p$ and to use the existence results already established in Boukrouche et al. (2020). Then we apply both compactness arguments and a fixed point method to prove the existence of a solution to our original fluid flow problem.