Quasilinear Differential Equations Research Articles

Uniform estimates for positive solutions of higher order quasilinear differential equations

For a higher order quasilinear differential equation, the existence of uniform estimates for positive solutions with common domain of definition is proved; these estimates depend on the estimates for the coefficients of the equation and do not depend on the coefficients themselves.

A class of singularly perturbed quasilinear differential equations with interior layers

A class of singularly perturbed quasilinear differential equations with discontinuous data is examined. In general, interior layers will appear in the solutions of problems from this class. A numerical method is constructed for this problem class, which involves an appropriate piecewise-uniform mesh. The method is shown to be a parameter-uniform numerical method with respect to the singular perturbation parameter. Numerical results are presented, which support the theoretical results.

On the well-posedness of the Cauchy problem for quasilinear differential equations of neutral type

This paper proves the theorems on the continuous dependence of solutions when perturbations of the initial moment, the initial vector, the initial functions, the matrix coefficients, and the nonlinear summand on the right-hand side are small in the Euclidean and integral topologies, respectively.

Approximate synthesis of optimal control over quasilinear stochastic differential equations with a small parameter and Poisson perturbations

The authors obtain successive approximations to the optimal control over a quasilinear system of stochastic functional-differential equations with Poisson disturbances and a quadratic quality functional.

On uniform estimates for solutions to quasi-linear differential equations

We obtain uniform estimates for positive solutions with the same domain to the equation $$y^{(n)} + \sum\limits_{i = 0}^{n - 1} {a_i (x)y^{(i)} } + p(x)\left| y \right|^{k - 1} y = 0$$ of even order n with k > 1 and continuous functions p(x) > 0 and a i (x). In the case where a 0(x) ≡ ⋯ ≡ a n−1(x) ≡ 0, the uniform estimates obtained depend on p = inf p(x) > 0 and are independent of the function p(x) itself.

A priori estimates, existence and non-existence for quasilinear cooperative elliptic systems

Let be a real number and let , , be a connected smooth domain. Consider the system of quasi-linear elliptic differential equations where , , and are real functions. Relations between the Liouville non-existence and a priori estimates and existence on bounded domains are studied. Under appropriate conditions, a variety of results on a priori estimates, existence and non-existence of positive solutions have been established.Bibliography: 11 titles.

Analytical-numerical investigation of a singular boundary value problem for a generalized Emden–Fowler equation

In this paper we are concerned about a singular boundary value problem for a quasilinear second-order ordinary differential equation, involving the one-dimensional p -laplacian. Asymptotic expansions of the one-parameter families of solutions, satisfying the prescribed boundary conditions, are obtained in the neighborhood of the singular points and this enables us to compute numerical solutions using stable shooting methods.

Implicit difference methods for quasilinear parabolic functional differential problems of the Dirichlet type

Classical solutions of quasilinear functional differential equations are approximated with solutions of implicit difference schemes. Proofs of convergence of the difference methods are based on a comparison technique. Nonlinear estimates of the Perron typ

On Positive Decaying Solutions of Quasilinear Ordinary Differential Equations with Singular Nonlinearities

We consider singular boundary value problems to quasilinear ordinary differential equations with singular nonlinearities. We give sufficient conditions for such problems to have a positive solution. Moreover, for a typical case we can show also the uniqueness of the solutions.

A finite difference method for quasi-linear and nonlinear differential functional parabolic equations with Dirichlet's condition

We deal with a finite difference method for a wide class of nonlinear, in particular strongly nonlinear or quasi-linear, second-order partial differential functional equations of parabolic type with Dirichlet's condition. The functional dependence is of t

Harnack estimates for quasi-linear degenerate parabolic differential equations

We establish the intrinsic Harnack inequality for non-negative solutions of a class of degenerate, quasilinear, parabolic equations, including equations of the p-Laplacian and porous medium type. It is shown that the classical Harnack estimate, while failing for degenerate parabolic equations, it continues to hold in a space-time geometry intrinsic to the degeneracy. The proof uses only measure-theoretical arguments, it reproduces the classical Moser theory, for non-degenerate equations, and it is novel even in that context. Holder estimates are derived as a consequence of the Harnack inequality. The results solve a long standing problem in the theory of degenerate parabolic equations.

Linearized maximum principle for neutral-type, variable-structure optimal problems with delays in controls

The authors state and study an optimal problem for variable-structure systems described by neutral-type quasilinear differential equations with discontinuous initial condition and incommensurable delays. The variation of the system structure means that in the process of motion, at a certain instant of time not known in advance, the object considered can pass from one law of motion to another, and, moreover, the initial condition for each of the subsequent states of the system depends on the state of the next to the last. The discontinuity of the initial condition means that at the initial instant of time, the value of the initial function and that of the trajectory do not coincide in general. The necessary optimality conditions are proved in the form of the linearized integral maximum principle for controls and initial functions and in the form of inequalities and equalities for the initial and final instants of structure change.

Asymptotic results for a class of fourth order quasilinear difference equations

The authors consider the fourth order quasilinear difference equation where α and β are positive constants and {p n } and {q n } are positive real sequences. They classify the nonoscillatory solutions according to their asymptotic behavior for large n and then give necessary and sufficient conditions for existence of solutions of these various types. The results are illustrated with examples.

Stability of solutions to delay differential equations with periodic coefficients of linear terms

Under study are the systems of quasilinear delay differential equations with periodic coefficients of linear terms. We establish sufficient conditions for the asymptotic stability of the zero solution, obtain estimates for solutions which characterize the decay rate at infinity, and find the attractor of the zero solution. Similar results are obtained for systems with parameters.

Oscillation of solutions of the systems of quasilinear hyperbolic equation under nonlinear boundary condition

Sufficient conditions are obtained for the oscillation of solutions of the systems of quasilinear hyperbolic differential equation with deviating arguments under nonlinear boundary condition.

On Existence of Nonoscillatory Solutions to Quasilinear Differential Equations

Abstract Sufficient conditions are established for the existence of nonoscillatory solutions to a quasilinear ordinary differential equation of higher order. For the equation with a positive potential, a criterion is established for the existence of nonoscillatory solutions with nonzero limit at infinity. In the case of even order, a criterion is obtained for all solutions at infinity to be oscillatory.

Uniform estimates of positive solutions to quasi-linear differential equations

One considers the differential inequality $$y^n + \sum\limits_{j = 0}^{n - 1} {a_j (x)y^{(j)} \geqslant p * \left| y \right|^k } ,$$ , where a j (x) are continuous functions, p* > 0, n ≥ 1, k > 1, and its special case $$r_n (x)\frac{d}{{dx}}\left( {...\frac{d}{{dx}}\left( {r_1 (x)\frac{d}{{dx}}(r_0 (x)y)} \right)...} \right) \geqslant \left| y \right|^k ,$$ , where all r j (x) are sufficiently smooth positive functions. Uniform estimates are obtained for solutions defined in the same domain.

Oscillations of solutions of second order quasilinear differential equations with impulses

Some Kamenev-type oscillation criteria are obtained for a second order quasilinear damped differential equation with impulses. These criteria generalize and improve some well-known results for second order differential equations with /and without impulses. In addition, new oscillation criteria are also obtained to generalize and improve known results. Two examples of applications are given to illustrate the theory.

A class of boundary traces for solutions of the equation [formula omitted

Our subject is the class U of all positive solutions of a semilinear equation L u = ψ ( u ) in E where L is a second order elliptic differential operator, E is a domain in R d and ψ belongs to a convex class Ψ of C 1 functions which contains functions ψ ( u ) = u α with α > 1 . A special role is played by a class U 0 of solutions which we call σ-moderate. A solution u is moderate if there exists h ⩾ u such that L h = 0 in E. We say that u ∈ U is σ-moderate if u is the limit of an increasing sequence of moderate solutions. In [E.B. Dynkin, S.E. Kuznetsov, Fine topology and fine trace on the boundary associated with a class of quasilinear differential equations, Comm. Pure Appl. Math. 51 (1998) 897–936] all σ-moderate solutions were classified by using their fine boundary traces. 2 2 It is known (see [B. Mselati, Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation, Mem. Amer. Math. Soc. 168 (798) (2004); E.B. Dynkin, Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 2004]) that U 0 = U in the case of the equation Δ u = u α with 1 < α ⩽ 2 in a bounded smooth domain. Therefore in this case the description of all solutions of L u = ψ ( u ) follows from the results on σ-moderate solutions. In [M. Marcus, L. Véron, The precise boundary trace of positive solutions of the equation Δ u = u q in the supercritical case, in: Perspectives in Nonlinear Partial Differential Equations, in honor of Haim Brezis, Contemp. Math., Amer. Math. Soc., Providence, RI, 2007, arxiv.org/math/0610102] Marcus and Véron introduced a different characteristic (called the precise trace) for solutions of the equation Δ u = u α with α > 1 in a bounded C 2 domain. In the present paper we develop a general scheme covering both approaches and we prove the equivalence, in a certain sense, of the fine and precise traces. An implication of this equivalence is a Wiener type criterion for vanishing of the Poisson kernel of the equation L u ( x ) = a ( x ) u ( x ) .

Oscillation criteria for second order quasi-linear neutral delay differential equations

In this paper, some new oscillation criteria for second order neutral nonlinear differential equation [ r ( t ) | ( x ( t ) + p ( t ) x ( ( σ ( t ) ) ) ) ′ | m - 1 ( x ( t ) + p ( t ) x ( ( σ ( t ) ) ) ) ′ ] ′ + q ( t ) f ( x ( τ ( t ) ) ) = 0 are established. New oscillation criteria are different from most known ones in the sense that they are based on a class of new functions Φ ( t , s , l ) defined in the sequel.