The Cauchy Problem for a Class of Semilinear Wave Equations with Piecewise Constants Data

ＴｈｅＣａｕｃｈｙＰｒｏｂｌｅｍｆｏｒａＣｌａｓｓｏｆＳｅｍｉｌｉｎｅａｒＷａｖｅＥｑｕａｔｉｏｎｓｗｉｔｈＰｉｅｃｅｗｉｓｅＣｏｎｓｔａｎｔｓＤａｔａＺｈａｎｇＹｏｎｇｑｉａｎ（张永前）（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ，ＦｕｄａｎＵｎ...     

Robin Boundary Value Problem for One-Dimensional Landau–Lifshitz Equations

In this paper, we are concerned with the existence and uniqueness of global smooth solution for the Robin boundary value problem of Landau-Lifshitz equations in one dimension when the boundary value depends on time t. Furthermore, by viscosity vanishing approach, we get the existence and uniqueness of the problem without Gilbert damping term when the boundary value is independent of t.     

Existence for Semilinear Wave Equations with Low Regularity

In this paper, we study how much regularity of initial data is needed to ensure existence of a local solution to the following semilinear wave equations utt-Δu=F(u, Du), u(0, x)=f(x)∈HS,(?)tu(0, x)=g(x)∈HS-1, where F is quadratic in Du with D = ((?)t,(?)x1,…,(?)xn). We proved that the range of s is s≥n 1/2 δ, respectively, withδ>1/4 if n = 2, andδ>0 if n = 3, andδ≥0 if n≥4. Which is consistent with Lindblad's counterexamples [3] for n = 3, and the main ingredient is the use of the Strichartz estimates and the refinement of these.     

Well–posedness for an Abstract Semilinear Integro–differential Fractional Problem

We consider an abstract second order semilinear integrodifferential equation involving fractional time derivatives of order between 1 and 2. Well–posedness is established under appropriate conditions on the initial data and the nonlinearity which is itself a fractional integral. These conditions will determine the exact underlying space where to look for solutions.     

The Cauchy Problem for Nonlinear Parabolic Equations with Lévy Laplacian

A solution to the Cauchy problem for a rather general class of nonlinear parabolic equations involving the infinite-dimensional Laplacian Δ_L of the form , where f is a real function defined on R³ is presented. Mathematics Subject Classifications (2000) 35R15, 46G05.     

One-Dimensional Parametric Determining form for the Two-Dimensional Navier–Stokes Equations

The evolution of a determining form for the 2D Navier–Stokes equations (NSE) which is an ODE on a space of trajectories is completely described. It is proved that at every stage of its evolution, the solution is a convex combination of the initial trajectory and a chosen, fixed steady state, with a dynamical convexity parameter \(\theta \), which will be called the characteristic determining parameter. That is, we show a separation of variables formula for the solution of the determining form. Moreover, for a given initial trajectory, the dynamics of the infinite-dimensional determining form are equivalent to those of the characteristic determining parameter \(\theta \) which is governed by a one-dimensional ODE. This one-dimensional ODE is used to show that if the solution to the determining form converges to the fixed state it does so no faster than \({\mathcal {O}}(\tau ^{-1/2})\), otherwise it converges to a projection of some other trajectory in the global attractor of the NSE, but no faster than \({\mathcal {O}}(\tau ^{-1})\), as \(\tau \rightarrow \infty \), where \(\tau \) is the evolutionary variable in determining form. The one-dimensional ODE is also exploited in computations which suggest that the one-sided convergence rate estimates are in fact achieved. The ODE is then modified to accelerate the convergence to an exponential rate. It is shown that the zeros of the scalar function that governs the dynamics of \(\theta \), which are called characteristic determining values, identify in a unique fashion the trajectories in the global attractor of the 2D NSE     

On the Suitable Weak Solutions for the Cauchy Problem of the Boussinesq Equations

ＯｎｔｈｅＳｕｉｔａｂｌｅＷｅａｋＳｏｌｕｔｉｏｎｓｆｏｒｔｈｅＣａｕｃｈｙＰｒｏｂｌｅｍｏｆｔｈｅＢｏｕｓｓｉｎｅｓｑＥｑｕａｔｉｏｎｓＧｕｏＢｏｌｉｎｇ（郭柏灵），ＹｕａｎＧｕａｎｇｗｅｉ（袁光伟）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＰｈｇｓｉ...     

Kamenev-type Oscillation Criteria for Semilinear Elliptic Differential Equations

Oscillation criteria for semilinear elliptic differential equations are obtained. The results are extensions of integral averaging technique of Kamenev. General means are employed to establish our results.     

Concentrations Problem for Solutions to Compressible Navier–Stokes Equations

Doklady Mathematics - A three-dimensional initial-boundary value problem for the isentropic equations of the dynamics of a viscous gas is considered. The concentration phenomenon is that, for...     

Localization Problem for General Filtration Equations

ＬｏｃａｌｉｚａｔｉｏｎＰｒｏｂｌｅｍｆｏｒＧｅｎｅｒａｌＦｉｌｔｒａｔｉｏｎＥｑｕａｔｉｏｎｓＹｕａｎＨｏｎｇｊｕｎ（袁洪君）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＪｌｉｎＵｎｉｖｅｒｓｉｔｙ，Ｃｈａｎｇｃｈｕｎ，１３００２３）＆（Ｄｅ...     

The Initial Boundary Value Problem for Symmetric Long Wave Equations with Nonhomogeneous Boundary Value

Ｔｈｅ　Ｉｎｉｔｉａｌ　Ｂｏｕｎｄａｒｙ　Ｖａｌｕｅ　Ｐｒｏｂｌｅｍ　ｆｏｒ　Ｓｙｍｍｅｔｒｉｃ　Ｌｏｎｇ　Ｗａｖｅ　Ｅｑｕａｔｉｏｎｓ　ｗｉｔｈ　Ｎｏｎｈｏｍｏｇｅｎｅｏｕｓ　Ｂｏｕｎｄａｒｙ　ＶａｌｕｅＭｉａｏＣｈｃｎｘｉａ（苗晨霞）（Ｉｎｓｔｉ...     

On the Cauchy Problem for a Generalized Emden–Fowler-Type Equation

Mathematical Notes -     

Low Regularity Exponential-Type Integrators for Semilinear Schrödinger Equations

We introduce low regularity exponential-type integrators for nonlinear Schrödinger equations for which first-order convergence only requires the boundedness of one additional derivative of the solution. More precisely, we will prove first-order convergence in \(H^r\) for solutions in \(H^{r+1}\) (with \(r > d/2\)) of the derived schemes. This allows us lower regularity assumptions on the data than for instance required for classical splitting or exponential integration schemes. For one-dimensional quadratic Schrödinger equations, we can even prove first-order convergence without any loss of regularity. Numerical experiments underline the favorable error behavior of the newly introduced exponential-type integrators for low regularity solutions compared to classical splitting and exponential integration schemes.     

Blow Up of Solutions to Semilinear Wave Equations with Critical Exponent in High Dimensions

In this paper, the author considers the Cauchy problem for semilinear wave equations with critical exponent in n≥4 space dimensions. Under some positivity conditions on the initial data, it is proved that there can be no global solutions no matter how small the initial data are.     

Sturm–Liouville Problem for Second Order Ordinary Differential Equations Across Resonance

This paper is concerned with the existence and uniqueness of solutions to the Sturm–Liouville boundary value problem across resonance. By using optimal control theory, we present some global optimality results about the unique solvability for the Sturm–Liouville problem.     

The Cauchy Problem for Defocusing Nonlinear Schrödinger Equations with Non-Vanishing Initial Data at Infinity

For rather general nonlinearities, we prove that defocusing nonlinear Schrödinger equations in ? ⁿ (n ≤ 4), with non-vanishing initial data at infinity u ₀, are globally well-posed in u ₀ + H ¹. The same result holds in an exterior domain in ? ⁿ , n = 2, 3.     

Blow－Up and Mass Concentration of Solutions to the Cauchy Problem for Nonlinear Schrodinger Equations

Ｂｌｏｗ－ＵｐａｎｄＭａｓｓＣｏｎｃｅｎｔｒａｔｉｏｎｏｆＳｏｌｕｔｉｏｎｓｔｏ　ｔｈｅＣａｕｃｈｙＰｒｏｂｌｅｍｆｏｒＮｏｎｌｉｎｅａｒＳｃｈｒｏｄｉｎｇｅｒＥｑｕａｔｉｏｎｓ秦玉明Ｂｌｏｗ－ＵｐａｎｄＭａｓｓＣｏｎｃｅｎｔｒａｔｉｏｎｏｆＳｏｌｕ...     

On S-Shaped Bifurcation Curves for a Class of Perturbed Semilinear Equations

By making use of bifurcation analysis and continuation method, the authors discuss the exact number of positive solutions for a class of perturbed equations. The nonlinearities concerned are the so-called convex-concave functions and their behaviors may be asymptotic sublinear or asymptotic linear. Moreover, precise global bifurcation diagrams are obtained.