Well-posedness and blow-up for a modified two-component Camassa–Holm equation

In this article, we consider a newly modified two-component Camassa–Holm equation. First, we establish the local well-posedness result, then we present a precise blow-up scenario. Afterwards, we derive a new conservation law, by which and the precise blow-up scenario we prove three blow-up results and a blow-up rate estimate result.     

Existence and nonexistence of blow-up solutions for a Schrödinger equation involving a nonlinear operator

By introducing a growth condition and using an iterative technique, we establish the results for the nonexistence and existence of positive entire blow-up solutions for a Schrödinger equation involving a nonlinear operator. Our main results improve and extend some existing works. In addition, we also give an example to illustrate our results.     

Invariant sets and solutions to the generalized thin film equation

The invariant sets and the solutions of the 1 2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set E0 = {u : ux = vxF(u), uy = vyF(u)}, where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the l l-dimensional nonlinear evolution equations.     

Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results

We are concerned with the following class of equations with exponential nonlinearities:

$$\begin{aligned} \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mathrm {in}~B_1\subset \mathbb {R}^2, \end{aligned}$$

which is related to the Tzitzéica equation. Here \(h_1, h_2\) are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue. In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions. In the second part of the paper we consider the Tzitzéica equation on compact surfaces: we start by proving a sharp Moser–Trudinger inequality related to this problem. Finally, we give a general existence result.

     

Global existence and blow-up phenomena for a periodic 2-component Camassa–Holm equation

We first establish local well-posedness for a periodic 2-component Camassa?CHolm equation. We then present two global existence results for strong solutions to the equation. We finally obtain several blow-up results and the blow-up rate of strong solutions to the equation.     

Settings and methods for global sensitivity analysis – a short guide

Global sensitivity analysis offers a set of tools tailored to the impact assessment of certain assumptions on a models output. A recent book on the topic covers those issues [1]. Given the limited space for discussing thoroughly any of those methods, we will next summarize the main conclusions that derive from the application of various global sensitivity analysis methods on chemical models [2], econometric studies [3] financial models [4] and composite indicators [5, 6]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a fourth-order quasilinear elliptic equation of concave–convex type

We consider a fourth-order quasilinear equation depending on a nonnegative parameter λ and with subcritical or critical growth. Such equation is equivalent to a Hamiltonian system and the main goal of this work is to prove the existence of at least two positive and infinitely many solutions for such equation when the parameter λ is positive and small enough. This work was supported by FAPESP grant #07/54872-8.     

Normalized solutions for a fourth-order Schr?dinger equation with a positive second-order dispersion coefficient

In this paper, we study normalized solutions to a fourth-order Schr¨odinger equation with a positive second-order dispersion coefficient in the mass supercritical regime. Unlike the well-studied case where the second-order term is zero or negative, the geometrical structure of the corresponding energy functional changes dramatically and this makes the solution set richer. Under suitable control of the second-order dispersion coefficient and mass, we find at least two radial normalized solutions,...     

Nonexistence of single blow-up solutions for a nonlinear Schrödinger equation involving critical Sobolev exponent

In this paper we shall consider the critical elliptic equation where and a(x) is a real continuous, non negative function, not identically zero. By using a local Pohozaev identity, we show that problem (0.1) does not admit a family of solutions which blows-up and concentrates as at some zero point x₀ of a(x) if the order of flatness of the function a(x) at x₀ is      

Mathematical analysis of a tumor invasion model—global existence and stability

This work studies an outstanding reaction–diffusion system modeling tumor invasion, with interactions among tumor tissue, acid concentration and normal tissue. This model has very different features from the models extensively studied in the mathematics literature. The most challenge issue for mathematical analysis of the present model is the existence of classical solution, since the diffusion of tumor tissue is influenced by the density of normal cells and diffusion degeneracy arises when normal cells are at the carrying capacity. A rigorous proof of global existence and uniqueness of classical solutions is presented. Moreover, we study global dynamics of the solution, and show asymptotic stability of the four possible constant equilibria under various scenarios.     

Global existence of small solutions for the fourth-order nonlinear Schrödinger equation

We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation

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where \(n=1,2\). We prove global existence of small solutions under the growth condition of \(f\left( u\right) \) satisfying \(\left| \partial _{u}^{j}f\left( u\right) \right| \le C\left| u\right| ^{p-j},\) where \(p>1+\frac{4}{n},0\le j\le 3\).

     

On a nonlocal problem for a fourth-order parabolic equation with the fractional Dzhrbashyan–Nersesyan operator

We study a nonlocal problem for a fractional partial differential equation with the Dzhrbashyan–Nersesyan fractional differentiation operator. By separation of variables, we prove a theorem on the existence and uniqueness of a regular solution of this problem.     

Blow-up and global solutions for a class of time fractional nonlinear reaction–diffusion equation with weakly spatial source

This paper investigates the blow-up of solutions for a time fractional nonlinear reaction–diffusion equation with weakly spatial source. We first derive two sufficient conditions under which the solutions may blow up in finite time. Then, we prove the existence of global solution when the initial data are small enough. Moreover, the long time behavior of bounded solutions will be analyzed.     

Local and global properties of solutions of quasilinear Hamilton–Jacobi equations

We study some properties of the solutions of (E) −_Δpu+|∇u|^q=0$-{\mathrm{\Delta }}_{p}u+{|\mathrm{\nabla }u|}^q=0$ in a domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, mostly when p≥q>p−1$p\ge q>p-1$. We give a universal a priori estimate of the gradient of the solutions with respect to the distance to the boundary. We give a full classification of the isolated singularities of the nonnegative solutions of (E), a partial classification of isolated singularities of the negative solutions. We prove a general removability result expressed in terms of some Bessel capacity of the removable set. We extend our estimates to equations on complete noncompact manifolds satisfying a lower bound estimate on the Ricci curvature, and derive some Liouville type theorems.