Multidimensional analog of the two-dimensional perron effect of sign change of characteristic exponents for infinitely differentiable differential systems

We obtain a general n-dimensional analog of the two-dimensional (partial) Perron effect of sign change of all arbitrarily prescribed negative characteristic exponents of an n-dimensional differential system of the linear approximation with infinitely differentiable bounded coefficients to the positive sign for the characteristic exponents of all nontrivial solutions of a nonlinear n-dimensional differential system with infinitely differentiable perturbations of arbitrary order m > 1 of smallness in a neighborhood of the origin and growth outside it. These positive exponents take n values distributed over n arbitrarily prescribed disjoint intervals and are realized on solutions issuing from nested subspaces R ¹ ? R ² ? ... ? R ⁿ .     

Existenz und Eindeutigkeit für Lösungen des Neumann—Tricomi-Problems im ℝ2

In a bounded simply-connected domainG \( \subseteq \) ?² a boundary value problem for a linear partial differential equation of second orderLu=f is studied. The operatorL is elliptic inG?{y>0}, parabolic forG?{y=0} and hyperbolic inG?{y<0}. The boundary value problem consists in findingu satisfyingLu=f inG, d _n u=φ on the elliptic part of the boundary ofG, u=ψ on the noncharacteristic part (which is not empty) of the hyperbolic part of the boundary ofG.d _n u denotes the conormal (with respect toL) derivative ofu. It is proved that the problem has a generalized solution in anL ²-weight space. Uniqueness is otained in the class of quasiregular solutions. In order to get the results apriori estimates are proved; theorems from functional analysis are used.     

Boundary value problems for the Fitzhugh-Nagumo equations

This paper is concerned with the study of the FitzHugh-Nagumo equations. These equations arise in mathematical biology as a model of the transmission of electrical impulses through a nerve axon; they are a simplified version of the Hodgkin-Huxley equations. The FitzHugh-Nagumo equations consist of a non-linear diffusion equation coupled to an ordinary differential equation. v_t = v_xx + f(v) ? u, u_t = σv ? γu. We study these equations with either Dirichlet or Neumann boundary conditions, proving local and global existence, and uniqueness of the solutions. Furthermore, we obtain L_∞ estimates for the solutions in terms of the L₁ norm of the boundary data, when the boundary data vanish after a finite time and the initial data are zero. These estimates allow us to prove exponential decay of the solutions.     

On well-posedness of semilinear parabolic and elliptic problems in the hyperbolic space

We investigate existence and uniqueness of solutions to semilinear parabolic and elliptic equations in bounded domains of the n-dimensional hyperbolic space (n?3). Lp→Lq estimates for the semigroup generated by the Laplace-Beltrami operator are obtained and then used to prove existence and uniqueness results for parabolic problems. Moreover, under proper assumptions on the nonlinear function, we establish uniqueness of positive classical solutions and nonuniqueness of singular solutions of the elliptic problem; furthermore, for the corresponding semilinear parabolic problem, nonuniqueness of weak solutions is stated.     

Large time behavior of entropy solutions to some hyperbolic system with dissipative structure

In this paper the large time behavior of the global L∞ entropy solutions to the hyperbolic system with dissipative structure is investigated. It is proved that as t →∞ the entropy solutions tend to a constant equilibrium state in L2 norm with exponential decay even when the initial values are arbitrarily large. As an illustration, a class of 2 × 2 system is studied.     

Hyperbolic Equations with Growing Coefficients in Unbounded Domains

Let Ω ? ? ⁿ , n?≥?2, be an unbounded domain with a smooth (possibly noncompact) star-shaped boundary Γ. For the first mixed problem for a hyperbolic equation with an unbounded coefficient with power growth at infinity, the large-time behavior of the solutions is studied. Estimates for the resolvent of the spectral problem are obtained for various values of the parameters.     

Moving boundary i.b.v.p.s for Schrödinger equations: Solution bounds and related results

The paper is concerned with i.b.v.p.s for Schrödinger equations, linear and nonlinear, in a straight line region with prescribed, moving boundaries, upon which (time-dependent) Dirichlet conditions are specified. Bounds, in terms of data, are obtained for the L₂ norm of the spatial derivative of the solutions, or for a measure related thereto: in the context of expanding boundaries, pointwise bounds for the solution may be inferred both in the linear case and in some nonlinear cases (e.g. the defocusing case). Asymptotic properties of the bounds for the aforementioned norm are discussed in the linear case. The methodology of the paper is based on a particular compact formula for the aforementioned norm of an arbitrary, complex-valued function whose values are assigned, as functions of time, on the assigned, moving boundaries of a straight line region. The application of the methodology to i.b.v.p.s for other p.d.e.s is discussed briefly.     

Norm estimates for the Alexander transforms of convex functions of order alpha

The hyperbolic sup norm of the pre-Schwarzian derivative of a locally univalent function on the unit disk measures the deviation of the function from similarities. We present sharp norm estimates for the Alexander transforms of convex functions of order α, 0?α<1.     

The mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems in the first quadrant

In this paper, we investigate the mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems with nonlinear boundary conditions on a half-unbounded domain . Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C¹ solutions with the bounded L¹∩L^∞ norm of the initial data as well as their derivatives and appropriate boundary condition. Based on the existence results of global classical solutions, we also prove that when t tends to infinity, the solutions approach a combination of C¹ travelling wave solutions. Under the appropriate assumptions of initial and boundary data, the results can be applied to the equation of time-like extremal surface in Minkowski space R^1+(1+n).     

Direct Lyapunov method for hyperbolic systems on the plane with time-periodic coefficients

In the framework of the direct Lyapunov method, we establish tests for the exponential stability of solutions of boundary value problems (the mixed problem and the Cauchy problem for hyperbolic systems of the class in question) in the L ₂-norm. In these tests, the conditions on the derivative of the Lyapunov functional along the trajectories of the system are weakened compared with the known results for the case of any smooth coefficients. The main result is illustrated by the example of the mixed problem for the telegraph system with small friction periodically switched on.     

L 3/2-estimates of vector fields with L 1 curl in a bounded domain

This paper establishes the estimates of L ^3/2 norm of the vector fields in a bounded domain with vanishing tangential component on the boundary, in terms of the L ¹ norm of the curl, the negative exponent Sobolev norm of the divergence, and on some quantities depending on the topology of the domain. As the similar proof we also obtain the estimates of L ^p norm of the vector fields in terms of the negative exponent Sobolev norms of the curl and divergence.     

A remark on the mean curvature of a graph-like hypersurface in hyperbolic space

In this paper we investigate the mean curvature H of a radial graph in hyperbolic space Hn+1. We obtain an integral inequality for H, and find that the lower limit of H at infinity is less than or equal to 1 and the upper limit of H at infinity is more than or equal to −1. As a byproduct we get a relation between the n-dimensional volume of a bounded domain in an n-dimensional hyperbolic space and the (n−1)-dimensional volume of its boundary. We also sharpen the main result of a paper by P.-A. Nitsche dealing with the existence and uniqueness of graph-like prescribed mean curvature hypersurfaces in hyperbolic space.     

Minkowski-Bouligand dimension of solutions of the one-dimensional p-Laplacian

The upper Minkowski-Bouligand dimension of the graph of continuous solutions, denoted by dim_My, is studied for a class of nonlinear one-dimensional p-Laplacian. Some sufficient conditions on the nonlinearities are given such that dim_My takes the prescribed fractional values. Next, a relation between dim_My and the order of growth for singular behaviour of L^p norm of derivatives of solutions is given. Finally, the change and stability of dim_My are considered by means of a double-parametric problem.     

Optimal error estimate and superconvergence of the DG method for first-order hyperbolic problems

We consider the original discontinuous Galerkin method for the first-order hyperbolic problems in d-dimensional space. We show that, when the method uses polynomials of degree k, the L₂-error estimate is of order k+1 provided the triangulation is made of rectangular elements satisfying certain conditions. Further, we show the O(h^2k+1)-order superconvergence for the error on average on some suitably chosen subdomains (including the whole domain) and their outflow faces. Moreover, we also establish a derivative recovery formula for the approximation of the convection directional derivative which is superconvergent with order k+1.     

A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation

The classical solution of the Dirichlet problem with a continuous boundary function for a linear elliptic equation with Hölder continuous coefficients and right-hand side satisfies the interior Schauder estimates describing the possible increase of the solution smoothness characteristics as the boundary is approached, namely, of the solution derivatives and their difference ratios in the corresponding Hölder norm. We prove similar assertions for the generalized solution with some other smoothness characteristics. In contrast to the interior Schauder estimates for classical solutions, our established estimates for the differential characteristics imply the continuity of the generalized solution in a sense natural for the problem (in the sense of (n-1)-dimensional continuity) up to the boundary of the domain in question. We state the global properties in terms of the boundedness of the integrals of the square of the difference between the solution values at different points with respect to especially normalized measures in a certain class.     

On linear systems with quasiperiodic coefficients and bounded solutions

For a discrete dynamical system ω _n =ω₀+αn, where a is a constant vector with rationally independent coordinates, on thes-dimensional torus Ω we consider the setL of its linear unitary extensionsx _n+1=A(ω₀+αn)x _n , whereA (Ω) is a continuous function on the torus Ω with values in the space ofm-dimensional unitary matrices. It is proved that linear extensions whose solutions are not almost periodic form a set of the second category inL (representable as an intersection of countably many everywhere dense open subsets). A similar assertion is true for systems of linear differential equations with quasiperiodic skew-symmetric matrices.     

Curvature flow of complete hypersurfaces in hyperbolic space

In this paper we continue our study of finding the curvature flow of complete hypersurfaces in hyperbolic space with a prescribed asymptotic boundary at infinity. Our main results are proved by deriving a priori global gradient estimates and C ² estimates.     

Some applications of reverse Hölder inequalities with a boundary integral

We obtain an increase in the exponent of integrability of the derivatives of solutions of two classes of boundary-value problems. We derive estimates of the corresponding norms of the solutions. For a class of quasilinear elliptic systems we establish an L_p-estimate of the gradient of the solutions of class W _m ¹ ,m > 1, p > m, of a boundary-value problem with nonzero condition on the conormal derivative. To solve Signorini's problem we obtain an L_p-estimate, p > 2,of the second derivatives of an L ₂-solution with a nonzero one-sided restriction on the conormal derivative. The proof of both results is based on the application of an reverse Hölder inequality with a surface integral established earlier by the author. Bibliography: 5 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 13–29.     

Lower dimensional invariant tori with prescribed frequency for nonlinear wave equation

In this paper, one-dimensional (1D) nonlinear wave equation utt−uxx+mu+u³=0, subject to Dirichlet boundary conditions is considered. We show that for each given m>0, and each prescribed integer b>1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, which correspond to b-dimensional invariant tori of an associated infinite-dimensional dynamical system. In particular, these Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.     

Existence of global L∞ solutions to a generalized n × n hyperbolic system of LeRoux type

In this article, we give the existence of global L^∞ bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of LeRoux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2 × 2 to n × n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v₁=0} is another difficulty. We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.