Stability of the solutions of the Cauchy problem for first-order quasilinear equations

In this paper we establish estimates of the stability of the solutions of the Cauchy problem for the multidimensional quasilinear equation \(u_t + (\varphi _i (u))_{x_i } = \psi (t, x, u)\) relative to variation of the function ?_i(u), ψ(t, x, u).     

Qualitative properties of solutions of the Neumann problem for a higher-order quasilinear parabolic equation

The property of localization of perturbations is proved for a solution of an initial boundary-value Neumann problem in a regionD=x, t>0, where is a region in Rⁿwith a noncompact boundary.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 45, No. 11, pp. 1571–1579, November, 1993.     

Stabilized approximate solutions of the inverse time problem for a parabolic evolution equation

The set of all solutions of a composite fuzzy relation equation of Sanchez (Inform. and Control30 (1976)), defined on finite spaces, is studied by determining and characterizing all the lower solutions of such an equation.     

Error estimation of approximate solutions to one inverse problem for a parabolic equation

This paper is devoted to solving the inverse boundary problem of the heat diagnostics by the projective regularization method. We obtain exact with respect to the order error estimates of the corresponding approximate solution.     

An inverse problem for a quasilinear parabolic equation

Sunto Si considera un problema parabolico sovraderminato in una sola variabile spaziale per l'operatore quasilineare definito da · a₃ o u. Tale operatore contiene un termine incognito a(k {0, 1, 2, 3})del quale si studia la dipendenza dalle condizioni iniziali e alla frontiera. Si determinano due classi, rispettivamente di dati e di soluzioni ammissibili, ed una coppia di metriche rispetto alle quali l'applicazione dati(u, a_k)è lipschitziana. Si mostra infine che tale applicazione si mantiene hölderiana quando le metriche ammesse per i dati sono soltanto del tipo L.

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Lavoro eseguito con contributo del Ministero della Pubblica Istruzione e nell'ambito del Gruppo G.N.A.F.A. del C.N.R.     

Periodic solutions of a quasilinear wave equation

We study the properties of wave operators satisfying the periodicity condition with respect to time and homogeneous boundary conditions of the third kind and of Dirichlet type. We prove the existence of a nontrivial periodic (in time) sine-Gordon solution with homogeneous boundary conditions of the third kind and of Dirichlet type. We obtain theorems on the existence of periodic solutions of a quasilinear wave equation with variable (in x) coefficients and a boundary condition of the third kind.     

Multiplicity of solutions for a fourth-order quasilinear nonhomogeneous equation

We consider a fourth-order quasilinear nonhomogeneous equation which is equivalent to a nonhomogeneous Hamiltonian system. The purpose of this work is to prove the existence of at least two solutions for such equation when a certain parameter is small enough. Furthermore, under an additional hypothesis on positiveness of the nonhomogeneous part we prove that our solutions are positive.     

Estimates for large solutions of a quasilinear elliptic equation

Let \(\Omega \) be a bounded smooth domain of \(R^{n}\). We study the asymptotic behaviour of the solutions to the equation \(\triangle u-|Du|^{q}=f(u)\) in \(\Omega , 1<q<2,\) which satisfy the boundary condition \(u(x)\rightarrow \infty \) as \(x\rightarrow \partial \Omega \). These solutions are called large or blowup solutions. Near the boundary we give lower and upper bounds for the ratio \(\psi (u)/\delta \), where \(\psi (u) = \int _{u}^{\infty }1/\sqrt{2F}dt\), \(F'=f\), \(\delta =dist(x,\partial \Omega )\) or for the ratio \(u/\delta ^{(2-q)/(1-q)}\). When in particular the ratio \(f/F^{q/2}\)is regular at infinity, we find again known results (Bandle and Giarrusso, in Adv Diff Equ 1, 133–150, 1996; Giarrusso, in Comptes Rendus de l’Acad Sci 331, 777–782 2000).     

On time decay estimates for solutions of the second mixed problem for a quasilinear parabolic equation

We obtain conditions for the existence of a global solution of the second mixed problem for a quasilinear parabolic equation in an unbounded domain. The estimates for the decay of a solution that depend on the geometry of the domain are established. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 25–32, January–March, 2008.     

Existence and uniqueness of unbounded entropy solutions of the Cauchy problem for first-order quasilinear conservation laws

We indicate conditions for the well-posedness of the Cauchy problem for a scalar quasilinear conservation law in the class of locally bounded functions. We construct examples showing that if these conditions are violated, then the Cauchy problem may fail to have a generalized entropy solution.     

Global smooth solutions for a quasilinear fractional evolution equation

The global existence of smooth solutions to a class of quasilinear fractional evolution equations is proved. The proofs are based on L^p(L^q) L^p(L^q) maximal regularity results for the corresponding linear equations.     

Neumann problem for a quasilinear elliptic equation in a varying domain

We investigate the Neumann problem for a nonlinear elliptic operator of Leray–Lions type in ${\Omega }^{(s)}=\Omega \backslash F^{(s)}$, $s=1,2,\dots$, where Ω is a domain in ${\mathbf{R}}^n$$(n?3)$, $F^{(s)}$ is a closed set located in the neighborhood of a $(n?1)$-dimensional manifold Γ lying inside Ω. We study the asymptotic behavior of $u^{(s)}$ as $s\to \infty$, when the set $F^{(s)}$ tends to Γ. To cite this article: M. Sango, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

The oblique boundary-value problem for a quasilinear parabolic equation

The solvability of the oblique boundary-value problem for quasilinear parabolic nondivergent equations with singularities is studied. Bibliography: 9 titles. To Olga Ladyzhenskaya on the occasion of the jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992. pp. 118–131. Translated by V. I. Ochkur.     

Closeness of solutions of Cauchy's problem for second-order parabolic equations

Mathematical Notes -     

A nonlocal problem for a first-order partial differential equation

In this paper we study a nonlocal problem for a first-order partial differential equation with an integral condition instead of the standard boundary one. We prove that the problem under consideration is uniquely solvable.     

On time-periodic solutions of a quasilinear wave equation

Existence and regularization theorems are obtained for generalized solutions of a quasilinear wave equation with variable coefficients and homogeneous Dirichlet boundary conditions. The nonlinear term either exhibits a power growth or satisfies a nonresonance condition.