31 research outputs found
Anisotropic parabolic equations with variable nonlinearity
We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional p(x, t)-Laplacian. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time L∞ bounds for the weak solutions
A compactness lemma of aubin type and its application to degenerate parabolic equations
Let Ω ⊂ Rⁿ be a regular domain and Φ(s) ∈ C loc (R) be a given function. If M⊂ L₂ (0, T ; W½ (Ω)) ∩ L ∞ (Ω × (0, T )) is bounded and the set {∂t Φ(v)|v ∈ M} is bounded in L₂ (0, T ; W-¹₂ (Ω)), then there is a sequence {vk} ∈ M such that vk ⇀ v ∈ L₂ (0,T ; W¹₂ (Ω)), and vk → v, Φ(vk) → Φ(v) a.e. in Ωτ = Ω × (0, T). This assertion is applied to prove solvability of the one-dimensional initial and boundary-value problem for a degenerate parabolic equation arising in the Buckley-Leverett model of two-phase filtration. We prove existence and uniqueness of a weak solution, establish the property of finite speed of propagation and construct a self-similar solutio
Solvability of a free-boundary problem describing the traffic flows
We study a mathematical model of the vehicle traffc on straight freeways, which describes the traffc flow by means of equations of one dimensional motion of the isobaric viscous gas. The corresponding free boundary problem is studied by means of introduction of Lagrangian coordinates, which render the free boundary stationary. It is proved that the equivalent problem posed in a time-independent domain admits unique local and global in time classical solutions. The proof of the local in time existence is performed with stan- dard methods, to prove the global in time existence the system is reduced to a system of two second-order quasilinear parabolic equationsyesBelgorod State National Research Universit
On a degenerate parabolic equation in filtration theory : Monotonicity and C infinity-regularity of interface
Anisotropic parabolic equations with variable nonlinearity
We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional p(x, t)-Laplacian. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time L∞ bounds for the weak solutions
A compactness lemma of aubin type and its application to degenerate parabolic equations
Let Ω ⊂ Rⁿ be a regular domain and Φ(s) ∈ C loc (R) be a given function. If M⊂ L₂ (0, T ; W½ (Ω)) ∩ L ∞ (Ω × (0, T )) is bounded and the set {∂t Φ(v)|v ∈ M} is bounded in L₂ (0, T ; W-¹₂ (Ω)), then there is a sequence {vk} ∈ M such that vk ⇀ v ∈ L₂ (0,T ; W¹₂ (Ω)), and vk → v, Φ(vk) → Φ(v) a.e. in Ωτ = Ω × (0, T). This assertion is applied to prove solvability of the one-dimensional initial and boundary-value problem for a degenerate parabolic equation arising in the Buckley-Leverett model of two-phase filtration. We prove existence and uniqueness of a weak solution, establish the property of finite speed of propagation and construct a self-similar solutio