Abstract
The initial-boundary value problem of a porous medium equation with a variable exponent is considered. Both the diffusion coefficient ax,t and the variable exponent px,t depend on the time variable t, and this makes the partial boundary value condition not be expressed as the usual Dirichlet boundary value condition. In other words, the partial boundary value condition matching up with the equation is based on a submanifold of ∂Ω×0,T. By this innovation, the stability of weak solutions is proved.
Highlights
Academic Editor: Genni Fragnelli e initial-boundary value problem of a porous medium equation with a variable exponent is considered
E initial-boundary value problem of a porous medium equation with a variable exponent is considered. Both the diffusion coefficient a(x, t) and the variable exponent p(x, t) depend on the time variable t, and this makes the partial boundary value condition not be expressed as the usual Dirichlet boundary value condition
The partial boundary value condition matching up with the equation is based on a submanifold of zΩ × (0, T)
Summary
U(x, t) is said to be a weak solution of equation (1) with the initial value (3) in the sense lim u(x, t) −. If u(x, t) satisfies (4) or (3) in the sense of the trace in addition, it is said to be a weak solution of the initial-boundary value problem of equation (1). If u0(x) ≥ 0 satisfies (16), equation (1) with initial value (3) has a nonnegative solution. Ω en, the initial-boundary value problem (1), (3), and (4) has a uniqueness solution. Let u(x, t) and v(x, t) be two solutions of equation (1) with the initial values u0(x) and v0(x), respectively, and with a partial boundary value condition u(x, t) v(x, t) 0, (x, t) ∈ Σ1 ∪ Σ2. At the end of this section, we would like to suggest that if m(x, t) m is a constant, condition (22) in eorem 3 is naturally true
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