Class Of Quasilinear Equations Research Articles

On the Numerical Solution of the Characteristic Problems for One Class of Quasilinear Equations

On the Numerical Solution of the Characteristic Problems for One Class of Quasilinear Equations

A Class of Quasilinear Equations with Hilfer Derivatives

A Class of Quasilinear Equations with Hilfer Derivatives

A Class of Quasilinear Equations with Distributed Gerasimov–Caputo Derivatives

Quasilinear equations in Banach spaces with distributed Gerasimov–Caputo fractional derivatives, which are defined by the Riemann–Stieltjes integrals, and with a linear closed operator A, are studied. The issues of unique solvability of the Cauchy problem to such equations are considered. Under the Lipschitz continuity condition in phase variables and two types of continuity over all variables of a nonlinear operator in the equation, we obtain two versions on a theorem on the nonlocal existence of a unique solution. Two similar versions of local unique solvability of the Cauchy problem are proved under the local Lipschitz continuity condition for the nonlinear operator. The general results are used for the study of an initial boundary value problem for a generalization of the nonlinear phase field system of equations with distributed derivatives with respect to time.

Existence of solutions for a class of quasilinear equations with vanishing potentials

We show the existence of positive solutions for a class of problems given by where , , and μ is a positive parameter. The potential V can vanish at infinity and f can be supercritical at the origin. We use the Mountain Pass Theorem and Del Pino & Felmer's arguments to overcome the lack of compactness.

A Class of Quasilinear Equations with Riemann–Liouville Derivatives and Bounded Operators

The existence and uniqueness of a local solution is proved for the incomplete Cauchy type problem to multi-term quasilinear fractional differential equations in Banach spaces with Riemann–Liouville derivatives and bounded operators at them. Nonlinearity in the equation is assumed to be Lipschitz continuous and dependent on lower order fractional derivatives, which orders have the same fractional part as the order of the highest fractional derivative. The obtained abstract result is applied to study a class of initial-boundary value problems to time-fractional order equations with polynomials of an elliptic self-adjoint differential operator with respect to spatial variables as linear operators at the time-fractional derivatives. The nonlinear operator in the considered partial differential equations is assumed to be smooth with respect to phase variables.

On a Class of Quasilinear Equations Involving Critical Exponential Growth and Concave Terms in $${\mathbb {R}}^N$$

On a Class of Quasilinear Equations Involving Critical Exponential Growth and Concave Terms in $${\mathbb {R}}^N$$

On the inverse variational problem for one class of quasilinear equations

The paper concerns the inverse problem of calculus of variations for one class of elliptic and hyperbolic quasilinear second order equations with two independent variables. The equations of this class have a rather wide range of applications, among which are modeling of a two-conductor transmission line, motion of a hyperelastic homogeneous rod whose cross-sectional area varies along the rod, vibration of a string, wave propagation in a bar of elastic–plastic material, and isentropic flows of a compressible gas with plane symmetry. A constructive solution of the problem in hand is given and the corresponding Lagrangians are explicitly constructed.

Well-posedness of quasilinear singular diffusion equations with boundary degeneracy

This paper concerns the well-posedness problems of a class ofquasilinear singular diffusion equations with boundary degeneracy.In addition to the possibledegeneracy and singularity likethe non-Newtonian filtration equations, the equations aredegenerate on a portion of the boundary due to the diffusion coefficients.The diffusion coefficients are anisotropic.For the well-posedness problems of such equations,the boundary condition may not be prescribed on the entire lateral boundary.The prescription of the boundary condition is determined bythe degeneracy of the diffusion coefficients along the outernormal direction of the spatial domain.The well-posedness problems of the equations are formulatedand the existence and uniqueness of weak solutions to the problems are proved.

Infinitely many solutions for a class of quasilinear equation with a combination of convex and concave terms

We consider the following quasilinear elliptic equation with convex and concave nonlinearities: \begin{equation*} -\Delta_p u-(\Delta_pu^2)u+V(x)|u|^{p-2}u=\lambda K(x) |u|^{q-2}u+\mu g(x,u),\quad \text{in }\mathbb{R}^N, \end{equation*} where $2\leq p< N$, $1< q< p$, $\lambda,\mu\in\mathbb{R}$, $V$ and $K$ are potential functions, and $g\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$ is a continuous function. Under some suitable conditions on $V,K$ and $g$, the existence of infinitely many solutions is established.

A Schechter type critical point result in annular conical domains of a Banach space and applications

Using Ekeland's variational principle we obtain a critical point theorem of Schechter type for extrema of a functional in an annular conical domain of a Banach space. The result can be seen as a variational analogue of Krasnoselskii's fixed point theorem in cones and can be applied for the existence, localization and multiplicity of the positive solutions of variational problems. The result is then applied to $p$-Laplace equations, where the geometric condition on the boundary of the annular conical domain is established via a weak Harnack type inequality given in terms of the energetic norm. This method can be applied also to other homogeneous operators in order to obtain existence, multiplicity or infinitely many solutions for certain classes of quasilinear equations.

EMBEDDING AND TRACE RESULTS FOR VARIABLE EXPONENT SOBOLEV AND MAZ'YA SPACES ON NON-SMOOTH DOMAINS

AbstractWe establish interior and trace embedding results for Sobolev functions on a class of bounded non-smooth domains. Also, we define the corresponding generalized Maz'ya spaces of variable exponent, and obtain embedding results similar as in the constant case. Some relations between the variable exponent Maz'ya spaces and the variable exponent Sobolev spaces are also achieved. At the end, we give an application of the previous results for the well-posedness of a class of quasi-linear equations with variable exponent.

Error estimates for optimal control problems of a class of quasilinear equations arising in variable viscosity fluid flow

We consider optimal control problems of quasilinear elliptic equations with gradient coefficients arising in variable viscosity fluid flow. The state equation is monotone and the controls are of distributed type. We prove that the control-to-state operator is twice Frechet differentiable for this class of equations. A finite element approximation is studied and an estimate of optimal order h is obtained for the control. The result makes use of the distributed structure of the controls, together with a regularity estimate for elliptic equations with Holder coefficients and a second order sufficient optimality condition. The paper ends with a numerical experiment, where the approximation order is computationally tested.

Computability aspects for 1st-order partial differential equations via characteristics

In this paper, it is shown that the method of characteristics can be used to compute local solutions of the boundary value problems for the first-order partial differential equations at feasible instances, under the natural definition of computability from the view point of application; but the maximal region of existence of a computable local solution may not be computable. It is also shown that the problem whether a boundary value problem has a global solution is not algorithmically decidable. The negative results retain even within the class of quasilinear equations defined by analytic computable functions over particularly simple domains (quasilinear equations are among the simplest first-order nonlinear partial differential equations). This fact shows that the algorithmic unsolvability is intrinsic.

Multiple sign changing solutions in a class of quasilinear equations

This paper deals with finding multiple sign-changing solutions of the following class of quasilinear problems: $$ \begin{cases} - (r^{\alpha} |u'(r)|^{\beta}u'(r))^{'} = \lambda~ r^{\gamma} f(u(r)),~~ 0 < r < R\\ u(R)= u'(0)=0 , \end{cases} $$ where $\alpha,$ $\beta$ and $\gamma$ are given real numbers, $\lambda > 0$ is a parameter, $f: {\bf R} \to {\bf R}$ is some continuous function and $0 < R < \infty$. A result on existence of infinitely many sign-changing solutions is obtained by considering a family of associated initial value problems which are solved through a shooting argument and a counting of zeroes.

A Class of Quasi-linear Equations in Coframe Gravity

We have shown recently that the gravity fieldphenomena can be described by a traceless part of thewave-type field equation. This is an essentiallynon-Einsteinian gravity model. It has an exactsphericallysymmetric static solution, that yields to theYilmaz-Rosen metric. This metric is very close to theSchwarzchild metric. The wave-type field equation cannotbe derived from a suitable variational principle by free variations, as was shown by Hehl and hiscollaborators. In the present work we are looking foranother field equation having the same exactspherically-symmetric static solution. Thedifferential-geometric structure on the manifold endowed with a smoothorthonormal coframe field is described by the scalarobjects of anholonomity and its exterior derivative. Weconstruct a list of the first and second order SO(1,3)-covariants (one- and two-indexedquantities) and a quasi-linear field equation with freeparameters. We fix a part of the parameters by acondition that the field equation is satisfied by aquasi-conformal coframe with a harmonic conformal function .Thus we obtain a wide class of field equations with asolution that yields the Majumdar-P apapetrou metricand, in particular, the Yilmaz-Rosen metric, that is viable in the framework of three classicaltests.

Existence of global weak solutions for a class of quasilinear equations describing Joule's heating

Existence of global weak solutions for a class of quasilinear equations describing Joule's heating

On the behavior of solutions of quasilinear elliptic equations of second order in unbounded domains

Analogues of the well known in the theory of analytic functions Phragmen-Lindeloff theorem are formulated for the solutions of a wide class of quasilinear equations of elliptic type. Examples are given which illustrate the sharpness of the obtained results for solutions of equations of the form div(|∇u|α−2∇u)=f(x, u), where the function f(x, u) is locally bounded in IRn+1,f(x, 0)=0,uf(x, u)⩾a¦u¦1+q,a>0,α>1,α-1>q⩾0, n>/2.

The existence of generalized solutions for a class of quasi-linear equations of mixed type

We consider equations in a bounded simply connected region G, where k( ~1) $0 for y $0 and G is bounded by the curves r,,, f-, , fz. Hereby r,, is a piecewise smooth curve lying in the half-plane y>O which intersects the line y=O in the points A( - 1,0) and B(0, 0). f, is a piecewise smooth curve through A in y < 0 which meets the characteristic of (1.1) issued from

General boundary value problem for quasilinear (A, $$\overrightarrow \beta$$ ) -elliptic equations

One gives the formulation of the general boundary value problem for a large class of quasilinear equations of the divergence form, admitting a fixed ellipticity degeneracy on an arbitrary subset of the domain of variation of the independent variables. One finds conditions for the existence and the uniqueness of generalized solutions of the indicated problem.

The conditions for there to be no global solutions of a class of quasilinear parabolic equations

Sufficient conditions are determined for the unboundedness (unsolvability ‘in the large’) of the solutions of boundary value problems for a class ofquasilinear equations and systems of equations of parabolic type, describing the propagation of heat in media with non-linear heat conduction and volume liberation of energy.