Existence of solutions to an initial Dirichlet problem of evolutional \( p(x) \)-Laplace equations

Abstract

The existence and uniqueness of weak solutions are studied to the initial Dirichlet problem of the equationwith 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 Hölder 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.

The one dimensional parabolic p(x)-Laplace equation

“…Consider the following quasilinear parabolic equation The case when p is constant was studied by a lot of authors and optimal results concerning this equation was obtained (see, for example, [5]). There recently appeared a large number of publications with non constant p, see [1][2][3][4]6,8,9,12] and the references therein. In [8] the multidimensional case was considered and it was proved that if p(x) > 0 is a measurable function, f = f (t, x, u) is C 1 function and u 0 ∈…”

Singular phenomena of solutions for nonlinear diffusion equations involving p(x)-Laplace operator and nonlinear sources

“…For the existence of solutions to Problem (1.1), we have the following theorem Theorem 2.1. [8,9] Suppose that Conditions (1.2) − (1.3) are fulfilled. Then for every…”

On a weak solution matching up with the double degenerate parabolic equation

“…In this paper, the double degenerate evolutionary p(x)-Laplacian equation u t = div b(x, t) ∇A(u) p(x)-2 ∇A(u) + f (x, t, u, ∇u), (x, t) ∈ Q T = Ω × (0, T), (1.1) is considered, in which Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, p(x) > 1 is a C 1 [1,2]), and has been widely studied [2][3][4][5][6][7][8][9][10][11][12][13][14][15] in recent decade. If b(x, t) = 1, p(x) = p > 1 is a constant, equation (1.1) is a generalization of the following polytropic infiltration equation:…”