Removability of isolated singularity for anisotropic parabolic equations with absorption

We study the problem of removability of isolated singularity for general quasilinear anisotropic parabolic equations with absorption. The precise conditions on the behaviour of the absorption term that ensure the non-existence of solutions with point singularities are established.     

Blow-up and global solutions for quasilinear parabolic equations with Neumann boundary conditions

Using the upper and lower solution techniques and Hopf's maximum principle, the sufficient conditions for the existence of blow-up positive solution and global positive solution are obtained for a class of quasilinear parabolic equations subject to Neumann boundary conditions. An upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also specified.     

Boundary value problem for a class of degenerate quasilinear parabolic equations with singularity

This paper is concerned with a class of quasilinear parabolic equations with singularity and arbitrary degeneracy. The existence and uniqueness of generalized solutions to a kind of boundary value problem is established.     

On quasilinear parabolic equations which admit global solutions for initial data with unrestricted growth

Three classes of quasilinear parabolic equations which have the common feature that their principal coefficients decay as the solution or its gradient blows up are studied. Long time existence of solutions for their Cauchy problems for initial data with arbitrary growth is established. Received September 9, 1999 / Accepted May 9, 2000 / Published online September 14, 2000     

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.     

Removability of isolated singularities for anisotropic elliptic equations with gradient absorption

We study the well-posedness of the third-order degenerate differential equation \(\left( {{P_3}} \right):\alpha {\left( {Mu} \right)^{\prime \prime \prime }}\left( t \right) + {\left( {Mu} \right)^{\prime \prime }}\left( t \right) = \beta Au\left( t \right) + f\left( t \right)\), (t ∈ [0, 2p]) with periodic boundary conditions \(Mu\left( 0 \right) = Mu\left( {2\pi } \right),\;Mu'\left( 0 \right) = Mu'\left( {2\pi } \right),\;Mu''\left( 0 \right) = Mu''\left( {2\pi } \right)\), in periodic Lebesgue–Bochner spaces L^p(T,X), periodic Besov spaces B_p,q^s(T,X) and periodic Triebel–Lizorkin spaces F_p,q^s(T,X), where A, B and M are closed linear operators on a Banach space X satisfying D(A) \( \cap \)D(B) ? D(M) and α, β, γ ∈ R. Using known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of (P₃) in the above three function spaces.     

Stability of solutions of quasilinear parabolic equations

We bound the difference between solutions u and v of ut=aΔu+divxf+h and vt=bΔv+divxg+k with initial data φ and ψ, respectively, by

     

Global solutions of abstract quasilinear parabolic equations

Local and global existence and uniqueness for strict solutions of abstract quasilinear parabolic equations are studied. Applications to quasilinear parabolic partial differential equations are also given.     

Transformation of approximate solutions of linear parabolic equations into asymptotic solutions of quasilinear parabolic equations

Moscow Institute of Electronics and Mathematics. Translated from Matematicheskie Zametki, Vol. 56, No. 6, pp. 122–126, December, 1994.     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

Nonlocal Dirichlet problem for linear parabolic equations with degeneration

In the spaces of classical functions with power weight, we prove the correct solvability of the Dirichlet problem for parabolic equations with nonlocal integral condition with respect to the time variable and an arbitrary power order of degeneration of coefficients with respect to the time and space variables. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 109–121, January, 2007.     

Blow-up of sign-changing solutions to quasilinear parabolic equations

We study the blow-up of sign-changing solutions to the Cauchy problem for quasilinear parabolic equations of arbitrary order. Our approach is based on H. Levine’s remarkable idea of constructing a concavity inequality for a negative power of a standard positive definite functional. Combining this with the nonlinear capacity method, which is based on the choice of optimal test functions, we find conditions for the blow-up of solutions to the problems under consideration.     

Initial-boundary-value problems for quasilinear degenerate hyperbolic equations with damping. Neumann problem

We study the behavior of the total mass of the solution of Neumann problem for a broad class of degenerate parabolic equations with damping in spaces with noncompact boundary. New critical indices for the investigated problem are determined. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 272–282, February, 2006.