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 authors of this paper study the Dirichlet problem of the following equationThe 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 u t = div(|Du| p−2 Du) + |u| q−1 u − λ|Du| l in R N with p > 2. We first prove that for 1 l p − 1, the solution exists at least for a short time; then for p 2 l p − 1, the existence and nonexistence of global (in time) solutions are studied in various situations.
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