The existence of weak solutions is studied to the initial Dirichlet problem of the equation u t = u div(|∇ u | p ( x )−2 ∇ u ), with inf p ( x ) > 2. We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.

In this paper, we study the initial-boundary value problem of porous medium equation u t = Δu m + h (t)u p in a cone D = (0,∞) × Ω, where h(t) ∼ t σ . Let ω 1 denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l 2+(n−2)l = ω 1. We prove that if \(m < p \leqslant m + \tfrac{{2(\sigma + 1)}} {{n + l}} + \sigma (m - 1) \) , then the problem has no global nonnegative solutions for any nonnegative u 0 unless u 0 = 0; if \(p > m + \tfrac{{2(\sigma + 1)}} {{n + l}} + \sigma (m - 1) \) , then the problem has global solutions for some u 0 ≥ 0.

The existence and uniqueness of weak solutions are studied to the initial Dirichlet problem of the equationu(t) = div(vertical bar del u vertical bar(p(x)-2)del u) + f(x. t. u).with inf p(x) > 2. The problems describe the motion of generalized Newtonian fluids which were studied by some other authors in which the exponent p was required to satisfy a logarithmic Holder continuity condition. The authors in this paper use a difference scheme to transform the parabolic problem to a sequence of elliptic problems and then obtain the existence of solutions with less constraint to p(x). The uniqueness is also proved. (C) 2012 Elsevier Masson SAS. All rights reserved.

In this paper, we study the initial-boundary value problem of the porous medium equation u t  = Δu m  + V(x)u p in a cone D = (0, ∞) × Ω, where V(x) ~ (1 + |x|) σ . Let ω 1 denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω and let l denote the positive root of l 2 + (n − 2)l = ω 1. We prove that if m ≤ p ≤ m + (2 + σ)/(n + l), then the problem has no global nonnegative solutions for any nonnegative u 0 unless u 0 = 0; if p > m + (2 + σ)/n, then the problem has global solutions for some u 0 ≥ 0.

This paper deals with the following Non-Newtonian polytropic filtration equation ut=div(|▽um|p-2▽um) possessing the extinction and positivity of the solution by comparison principle and fundamental solution for m0,p1 under initial value u0(x)∈L1(Ω) and zero boundary value conditions.

This paper deals with the following Non-Newtonian polytropic filtration equation ut=div(|▽um|p-2▽um) possessing the extinction and positivity of the solution by comparison principle and fundamental solution for m>0,p>1 under initial value u0(x)∈L1(Ω) and zero boundary value conditions.

In this paper, we study a generalized thin film equation which is relevant to capillary driven flows of thin films of power-law fluids. We prove that the generalized thin film equation in dimension d >= 2 has a unique source type radial self-similar nonnegative solution if 0 < n < 2p-1 and has no solution of this type if n >= 2p-1.

In this paper, the authors consider the limiting problem of the drift-diffusion-Poisson model for semiconductors. Different from previous papers, the model considered involve some special doping profiles D which have the property that the function is allowed to have a jump-discontinuity and sign changing property but D 2 is required to be Lipschitz continuous. The existence, uniqueness and large-time asymptotic behavior of the global (in time) solutions are given.

For u0,1u0∈Lloc1(RN), the author studies the existence of a kind of weak solution to the Cauchy problemut=div(um−1Du),inRN×(0,T],u(x,0)=u0(x)⩾0,inRN, where m<0 is a constant. The uniqueness and regularity of solutions are also discussed.

The authors of this paper study the Dirichlet problem of the following equationut−div(|u|ν(x,t)∇u)=f−|u|p(x,t)−1u. The existence and uniqueness of weak solutions are proven. Also, the properties of the solutions are studied which include the property of finite speed of propagation of disturbances, localization property and the property of vanishing at a finite time etc.

In this paper, the authors study the equation ut=div(|Du|p−2Du)+|u|q−1u−λ|Du|l in RN with p>2. We first prove that for 1⩽l⩽p−1, the solution exists at least for a short time; then for p2⩽l⩽p−1, the existence and nonexistence of global (in time) solutions are studied in various situations.

The aims of this paper are to discuss the extinction and positivity for the solution of the initial boundary value problem and Cauchy problem of u t = div(|∇u m | p−2∇u m ). It is proved that the weak solution will be extinct for 1 < p ≤ 1 + 1/m and will be positive for p > 1 + 1/m for large t, where m > 0.

For the initial value problem of the parabolic Monge–Ampère equation VtVxx+rxVxVxx-θVx2=0, (x,t)∈R×[0,T) arising from the optimal investment of mathematical finance theory, we establish the existence of solutions, whose application is also given. Here the initial function is unbounded, and a special property is required for the solution to satisfy.

The aims of this paper are to discuss extinction and positivity for the evolution p-Laplacian equation ∂u∂t=∂∂x(|∂u∂x|p−2∂u∂x) with initial value u0(x)∈L1(Ω) and zero boundary value conditions, where p>1. In particular, the necessary and sufficient condition for extinction is obtained.

The author of [1] raised an optimal investment problem in time interval [0, T], in which the financial market is characterized by the parameters r, b, σ, the attitude of the investor to the risk versus the gain at the final time is described by a utility function g(y), the purpose is to find out an optimal portfolio to maximize the profit of the investor. To this end, in [1] the following initial value problem is derived:

The aim of this paper is to discuss the extinction and positivity for the evolution P-Laplacian equation [formula] with p > 1. In particular, the necessary and sufficient condition for extinction is obtained.

From the theory of optimal investment in mathematical finance, the following initial valueproblem for a parabolic Monge-Ampère equation was derived in [1]:

In this paper, we study the generalized solution of the first initial boundary value problem for the parabolic Monge–Ampère equation −utdetD2u=f(x,t) in Q=Ω×(0,T], u=ϕ(x,t) on ∂pQ. We first get the Hölder continuity of the generalized solution in t, and then obtain the generalized solution in case f(x,t)=0, which improve the assumptions in the existence of generalized solution in (J. Partial Differential Equations 14(2) (2001) 149).

The present paper covers the existence and uniqueness of classical solutions to the third initial and boundary value problem for the equation of parabolic Monge-Ampere type and a structure condition in reference [1] is improved.