Matrix inverse problem and its optimal approximation problem for R-skew symmetric matrices

Let R ∈ C^n×n be a nontrivial involution, i.e., R² = I and R ≠ ±I. A matrix A ∈ C^n×n is called R-skew symmetric if RAR = −A. The least-squares solutions of the matrix inverse problem for R-skew symmetric matrices with R∗ = R are firstly derived, then the solvability conditions and the solutions of the matrix inverse problem for R-skew symmetric matrices with R∗ = R are given. The solutions of the corresponding optimal approximation problem with R∗ = R for R-skew symmetric matrices are also derived. At last an algorithm for the optimal approximation problem is given. It can be seen that we extend our previous results [G.X. Huang, F. Yin, Matrix inverse problem and its optimal approximation problem for R-symmetric matrices, Appl. Math. Comput. 189 (2007) 482-489] and the results proposed by Zhou et al. [F.Z. Zhou, L. Zhang, X.Y. Hu, Least-square solutions for inverse problem of centrosymmetric matrices, Comput. Math. Appl. 45 (2003) 1581-1589].     

Convergence to strong nonlinear diffusion waves for solutions to p-system with damping

In this paper we consider the so-called p-system with linear damping, and we will prove an optimal decay estimates without any smallness conditions on the initial error. More precisely, if we restrict the initial data (V₀,U₀) in the space H³(R⁺)∩L1,γ(R⁺)×H²(R⁺)∩L1,γ(R⁺), then we can derive faster decay estimates than those given in [P. Marcati, M. Mei, B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech. 7 (2) (2005) 224-240; H. Zhao, Convergence to strong nonlinear diffusion waves for solutions of p-system with damping, J. Differential Equations 174 (1) (2001) 200-236] and [M. Jian, C. Zhu, Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant, J. Differential Equations 246 (1) (2009) 50-77].     

Porous Medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative

The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)~(-σ)u) = f, 0 σ 1/2.This paper poses the problem over {t ∈ R~+, x ∈ R~n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.     

Blowing up boundary conditions for a nonlocal nonlinear diffusion equation in several space dimensions

We analyze boundary value problems prescribing Dirichlet or Neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation in a bounded smooth domain Ω∈R^N with N≥1. First, we prove existence and uniqueness of solutions and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

The left and right inverse eigenvalue problems of generalized reflexive and anti-reflexive matrices

Let n×n complex matrices R and S be nontrivial generalized reflection matrices, i.e., R^∗=R=R⁻¹≠±I_n, S^∗=S=S⁻¹≠±I_n. A complex matrix A with order n is said to be a generalized reflexive (or anti-reflexive ) matrix, if RAS=A (or RAS=−A). In this paper, the solvability conditions of the left and right inverse eigenvalue problems for generalized reflexive and anti-reflexive matrices are derived, and the general solutions are also given. In addition, the associated approximation solutions in the solution sets of the above problems are provided. The results in present paper extend some recent conclusions.     

The properties of orthogonal matrix-valued wavelet packets in higher dimensions

In the paper matrix-valued multiresolution analysis and matrix-valued wavelet packets of spaceL ²(R ⁿ ,C ^{s x s}) are introduced. A procedure for constructing a class of matrix-valued wavelet packets in higher dimensions is proposed. The properties for the matrix-valued multivariate wavelet packets are investigated by using integral transform, algebra theory and operator theory. Finally, a new orthonormal basis ofL ²(R ⁿ ,C ^{s x s}) is derived from the orthogonal multivariate matrix-valued wavelet packets.     

Solving deconvolution type problems by wavelet decomposition methods

The paper considers an inverse problem associated with equations of the form Kf = g, where K is a convolution-type operator. The aim is to find a solution f for given function g. We construct approximate solutions by applying a wavelet basis that is well adapted to this problem. For this basis we calculate the elementary solutions that are the approximate preimages of the wavelets. The solution for the inverse problem is then constructed as an appropriate finite linear combination of the elementary solutions. Under certain assumptions we estimate the approximation error and discuss the advantages of the proposed scheme.     

The class of solenoidal planar-helical vector fields

The class of solenoidal vector fields whose lines lie in planes parallel to R ² is constructed by the method of mappings. This class exhausts the set of all smooth planarhelical solutions of Gromeka’s problem in some domain D ? R ³. In the case of domains D with cylindrical boundaries whose generators are orthogonal to R ², it is shown that the choice of a specific solution from the constructed class is reduced to the Dirichlet problem with respect to two functions that are harmonic conjugates in D ² = D∩R ²; i.e., Gromeka’s nonlinear problem is reduced to linear boundary value problems. As an example, a specific solution of the problem for an axisymmetric layer is presented. The solution is based on solving Dirichlet problems in the form of series uniformly convergent in \(\bar D^2\) in terms of wavelet systems that form bases of various spaces of functions harmonic in D ².     

The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system

The blow-up solutions of the Cauchy problem for the Davey-Stewartson system, which is a model equation in the theory of shallow water waves, are investigated. Firstly, the existence of the ground state for the system derives the best constant of a Gagliardo-Nirenberg type inequality and the variational character of the ground state. Secondly, the blow-up threshold of the Davey-Stewartson system is developed in R³. Thirdly, the mass concentration is established for all the blow-up solutions of the system in R². Finally, the existence of the minimal blow-up solutions in R² is constructed by using the pseudo-conformal invariance. The profile of the minimal blow-up solutions as t→T (blow-up time) is in detail investigated in terms of the ground state.     

Lifespan Theorem for simple constrained surface diffusion flows

We consider closed immersed hypersurfaces in R³ and R⁴ evolving by a special class of constrained surface diffusion flows. This class of constrained flows includes the classical surface diffusion flow. In this paper we present a Lifespan Theorem for these flows, which gives a positive lower bound on the time for which a smooth solution exists, and a small upper bound on the total curvature during this time. The hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature. This turns out to be a deep property of the initial manifold, as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in L² for M² immersed in R³ and additionally on the concentration in L³ for M³ immersed in R⁴. This is stronger than a previous result on a different class of constrained surface diffusion flows, as here we obtain an improved lower bound on maximal time, a better estimate during this period, and eliminate any assumption on the area of the evolving hypersurface.     

A study on compactly supported orthogonal vector-valued wavelets and wavelet packets

In this paper, vector-valued multiresolution analysis and orthogonal vector-valued wavelets are introduced. The definition for orthogonal vector-valued wavelet packets is proposed. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is derived by means of paraunitary vector filter bank theory. An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented. The properties of the vector-valued wavelet packets are investigated by using operator theory and algebra theory. In particular, it is shown how to construct various orthonormal bases of L²(R, C^s) from the orthogonal vector-valued wavelet packets.     

Wavelet packets associated with nonuniform multiresolution analyses

A multiresolution analysis was defined by Gabardo and Nashed for which the translation set is a discrete set which is not a group. We construct the associated wavelet packets for such an MRA. Further, from the collection of dilations and translations of the wavelet packets, we characterize the subcollections which form orthonormal bases for L²(R).     

Reconstruction of two time independent coefficients in an inverse problem for a phase field system

In this paper we present stability results concerning the inverse problem of determining two time independent coefficients for a phase field system in a bounded domain Ω⊂Rⁿ for the dimension n≤3 with a single observation on a subdomain ω?Ω and the Sobolev norm of certain partial derivatives of the solutions at a fixed positive time θ∈(0,T) over the whole spatial domain. The proof of these results relies on an appropriate Carleman estimate for the phase field system.     

Weighted pseudo-almost periodic solutions of a class of abstract differential equations

For abstract linear functional differential equations with a weighted pseudo-almost periodic forcing term, we prove that the existence of a bounded solution on R⁺ implies the existence of a weighted pseudo-almost periodic solution. Our results extend the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations. To illustrate the results, we consider the Lotka-Volterra model with diffusion.     

Reverse time diffusions

The paper considers a diffusion evolving in $\text{R}$ⁿ. The stochastic differential equations giving the same process, but with the time parameter evolving in the negative direction, are obtained under a certain integrability hypothesis when the diffusion has a density function on a time varying submanifold of $\text{R}$ⁿ.     

Trivial stationary solutions to the Kuramoto-Sivashinsky and certain nonlinear elliptic equations

We show that the only locally integrable stationary solutions to the integrated Kuramoto-Sivashinsky equation in R and R² are the trivial constant solutions. We extend our technique and prove similar results to other nonlinear elliptic problems in RN.     

Long time behavior of heat kernels of operators with unbounded drift terms

Given a second-order elliptic operator on Rd, with bounded diffusion coefficients and unbounded drift, which is the generator of a strongly continuous semigroup on L²(Rd) represented by an integral, we study the time behavior of the integral kernel and prove estimates on its decay at infinity. If the diffusion coefficients are symmetric, a local lower estimate is also proved.     

Positive solutions of reaction diffusion equations with super-linear absorption: Universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions

Consider classical solutions u∈C²(Rⁿ×(0,∞))∩C(Rⁿ×[0,∞)) to the parabolic reaction diffusion equation

     

Oscillatory radial solutions for subcritical biharmonic equations

It is well known that the biharmonic equation Δ²u=u|u|^p−1 with p∈(1,∞) has positive solutions on Rn if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on Rn in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball.