The nonlinear diffusion equation of the ideal barotropic gas through a porous medium

Definition 1.1

A locally integral bounded function u(x, t) is said to be the weak solution of problem (2)-(3)-(4) if :

1. u ∈ L^∞(Q_T), |u|γ(x,t)2∇u∈L2(0,T;L2(Ω)),ut∈L2(0,T;H−1(Ω)).

2. The boundary condition (4) is satisfied in the sense of trace.

3. For any test function ζ(x, t) satisfying the conditions

and for 0 ≤ t₁ ≤ t₂ ≤ T, it holds

Let γ(x, t) be a appropriately smooth function in Q_T. If

Antontsev-Shmarev [1] had proved that the problem (2)-(3)-(4) has at least one weak solution in the sense of Definition 1.1. The solution is bounded and satisfies the estimate ||u||_∞,QT ≤ K(T) with the constant K(T) from condition (7).

In the present work, we limit ourselves to the study of the following equation

where ρ(x) = dist(x, ∂Ω), α > 0.

If α = 0 and γ(x, t) ≡ m = const, equation (8) is the usual porous medium equation

Also, when m = 1, f = 0, equation (9) is regarded as a nonlinear heat equation

which has been studied in many well-known monographs or textbooks, for examples one can refer to [2-7], where a wide spectrum of methods is used. The function k(u) has the meaning of nonlinear thermal conductivity, which depends on the temperature u = u(x, t). Meanwhile, if 0 > m > −1, equation (9) is called a fast diffusion equation. The name “fast diffusion” is related to the fact that since the heat conductivity is unbounded in the unperturbed (zero temperature) background, the heat propagates from warm regions into cold ones much faster than it propagates in the case of constant (m = 0 in (9)) heat conductivity, and even faster than in the case m > 0, in which the speed of propagation of perturbations is finite.

Definition 1.2

A function u(x, t) is said to be the weak solution of equation (8) with the initial value (3), if

for any function ϕ ∈ C¹(Q_T), ϕ |_t=T = 0, ϕ |_∂Ω = 0, and the initial value (3) is satisfied in the sense of that

The main results in our paper are the following theorems.

Theorem 1.3

Suppose that f(x, t) is a smooth function, u₀(x) satisfies (14). If α > 0, γ(x, t) ∈ L^∞(Q_T), ∇γ ∈ L¹(Q_T), γ⁻ > −1, then equation (8) with initial value (3) has a solution.

Theorem 1.4

If 0 < α, γ⁻ ≥ 1, u₀(x) and υ₀(x) satisfy (14). Let u, υ be two solutions of equation (8) with the initial values u₀(x), υ₀(x) respectively, and

Then

Theorem 1.5

If 0 < α, γ⁻ > −1, γ(x, t) ∈ C¹(Q_T), u₀(x) and υ₀(x) satisfy (14). Let u, υ be two solutions of equation (8) with the initial values u₀(x), υ₀(x) respectively, and

Then the stability (19) is true.

Corollary 1.6

If 0 < α, γ(x, t) ≡ γ > −1, u₀(x) and υ₀(x) satisfy (14). Let u, υ be two solutions of equation (8) with the initial values u₀(x), υ₀(x) respectively. Then the stability (19) is true.

Roughly speaking, we may conjecture only if α > 0, the stability (19) may be true without the condition (18) or (20). Theorem 1.4 and Theorem 1.5 have verified the fact partly. Recently, the author [12] had studied the equation

with α > 0, and had shown that the uniqueness of the solutions of equation (20) is true without any boundary value condition. Meanwhile, Jir̆i̍ Benedikt et.al [13, 14] had studied the equation

with 0 < α < 1, and shown that the uniqueness of the solutions of equation (22) is not true. From the short comment, one can see that the degeneracy of the coefficient ρ^α plays an important role in the well-posedness of the solutions, it even can eliminate the action from the source term f(u,x,t).

The paper is arranged as follows. In the first section, we give a brief introduction and narrate the main results. In the second section, we prove the existence. In the third section, we prove the stability of the solutions.

Proof of Theorem 1.3

Let u₀ ∈ L^∞(Ω) satisfy (14). Consider the regularized problem of equation (8)

with the initial-boundary value condition (3)-(4). Here a(ɛ,u,x,t) = (ρ^α + ɛ)(ɛ + |u_ɛ|)^{γ(x, t)}. Then

Similar to [1], by Schauder Fixed Point Theorem, we know that the regularized problem has a solution u_ɛ in the sense of Definition 1.1.

Proof of Theorem 1.4

For a small positive constant λ > 0, let

Now, if u₀ and υ₀ only satisfy (14), let u, υ be two solutions of equation (8) with the initial values u₀, υ₀ respectively. Then

By multiplying (38) by φ_λ(x)S_η(u − υ), and integrating it over Ω, we have

We now calculate the terms of (39) as follows.

Since

where min{|υ|, |u|} ≤ ζ ≤ max{|υ|, |u|}, by |Sη′(u−υ)|≤c|u−υ|,γ−≥1,

due to that the condition (18)

Using the dominated convergence theorem, by limη→0Sη′(s)s=0, we have

At the same time,

then

By

after letting λ → 0, let η → 0 in (39). Then