Abstract
Existence and nonexistence of global solutions for doubly nonlinear diffusion equations with logarithmic nonlinearity
Highlights
In this paper, we will study the following doubly nonlinear diffusion equations with logarithmic nonlinearity ut − ∆p u(m−1)= fq (u), x ∈ Ω, t > 0, u(x, t) = 0, x ∈ ∂Ω, t > 0, (1.1)u(x, 0) = u0(x), x ∈ Ω, where Ω ⊂ Rn (n ≥ 1) is an open bounded domain with smooth boundary ∂Ω, u(m−1) := |u|m−2 u, ∆p (u) := div (∇u)(p−1) the usual p-Laplacian operators and fq is of the form of logarithmic term fq(s) = s(q−1) log |s|.Let us consider the following equation which is so-called doubly nonlinear parabolic equations ut − ∆p u(m−1) = f (u), (1.2)N
U(x, 0) = u0(x), x ∈ Ω, where Ω ⊂ Rn (n ≥ 1) is an open bounded domain with smooth boundary ∂Ω, u(m−1) := |u|m−2 u, ∆p (u) := div (∇u)(p−1) the usual p-Laplacian operators and fq is of the form of logarithmic term fq(s) = s(q−1) log |s|
In [27], Payne and Sattinger developed the potential well method which is introduced by Lions [20] and Sattinger [30] to study the existence and nonexistence of global weak solutions to heat and wave equations with power like nonlinearity under condition of positive initial energy
Summary
We will study the following doubly nonlinear diffusion equations with logarithmic nonlinearity ut − ∆p u(m−1). In [27], Payne and Sattinger developed the potential well method which is introduced by Lions [20] and Sattinger [30] to study the existence and nonexistence of global weak solutions to heat and wave equations with power like nonlinearity under condition of positive initial energy. In the same spirit with previous works, we utilize the potential well method to study the existence and nonexistence of global weak solutions to (1.2) with logarithmic nonlinearities fq(u) = (u)(q−1) log |u|, q > 2 and initial value u0(m−1) belonging to Sobolev space W01,p (Ω). The rest of this paper is organized as follows: Section 2 devotes to preliminaries in which we establish some properties of stationary problem associated to (1.7) and introduce the stable sets (potential well) and unstable sets as well as its properties; Section 3 states main results of this paper and their proofs are presented in the remaining sections
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