In this paper, we consider the problem (θ(x,U))_t=(K(x,U)U_x)_x-(K(x,U))_x (x,t)∈G_T (θ(x,U)V(x,t))_t=(DθV_x)_x+(V(KU_x-K))_x,(x,t)∈G_T, u(x,0)=u_0(x),V(x,0),(x,0)=V_0(x),0≤x≤2, U(0,t)=h_0(t),U(2,t)=h_2(t),0≤t...In this paper, we consider the problem (θ(x,U))_t=(K(x,U)U_x)_x-(K(x,U))_x (x,t)∈G_T (θ(x,U)V(x,t))_t=(DθV_x)_x+(V(KU_x-K))_x,(x,t)∈G_T, u(x,0)=u_0(x),V(x,0),(x,0)=V_0(x),0≤x≤2, U(0,t)=h_0(t),U(2,t)=h_2(t),0≤t≤T, V(0,t)=g_0(t),V(2,t)=g_2(t),0≤t≤T. Where, θ(x,U)=θ_1(x,U) when (x,t)∈D_1={0≤x<1,0≤t≤T};θ(x,U)=θ_2(x,U),(x,t)∈D_2={1<x≤2,0≤t≤T}.K(x,U)=K_i(x,U),(x,t)∈D_i. θ_i, K_i are the Moisture content and hy draulic conductivity of porous Media on D_i respectively. V be the the concentration of solute in the fluid. In addition we also require that U, V, (K(x,U)U_x-1) and DθV_x+V(KU_x-K) are continu ous at x=1. We prove the exisence, uniqueness and large time behavior of the problem by the method of reg ularization.展开更多
In this paper,we consider the degenerate diffusion problem where Ω R^N is an open bounded domain with lipschta boundary Ω9. ψ(u) ∈ c^2[0,∞) ,ψ(s)>O,ψ′(s) >0, ψ′(s)≥0 ,whens>0;ψ(0)= 0,ψ′(0))≥ 0....In this paper,we consider the degenerate diffusion problem where Ω R^N is an open bounded domain with lipschta boundary Ω9. ψ(u) ∈ c^2[0,∞) ,ψ(s)>O,ψ′(s) >0, ψ′(s)≥0 ,whens>0;ψ(0)= 0,ψ′(0))≥ 0. u_0(x)∈ H_0~1(Ω) ∩ C^o(),u_o(x)≥0.g(s) is locally lipschitz continuous on [0,∞), g(0)=0. There exist a constant K,K> maxu_o(x), such that g(K) <0. |g(s)|/ψ(s)≤ M_o when 0≤s≤K ,where M_0 is a constant. We prove the existence and localization phenomena of weak solution of above problem. Under some additional conditions,we prove th uniqueness,contiouous and asymptotic behavier of weak solution.展开更多
文摘In this paper, we consider the problem (θ(x,U))_t=(K(x,U)U_x)_x-(K(x,U))_x (x,t)∈G_T (θ(x,U)V(x,t))_t=(DθV_x)_x+(V(KU_x-K))_x,(x,t)∈G_T, u(x,0)=u_0(x),V(x,0),(x,0)=V_0(x),0≤x≤2, U(0,t)=h_0(t),U(2,t)=h_2(t),0≤t≤T, V(0,t)=g_0(t),V(2,t)=g_2(t),0≤t≤T. Where, θ(x,U)=θ_1(x,U) when (x,t)∈D_1={0≤x<1,0≤t≤T};θ(x,U)=θ_2(x,U),(x,t)∈D_2={1<x≤2,0≤t≤T}.K(x,U)=K_i(x,U),(x,t)∈D_i. θ_i, K_i are the Moisture content and hy draulic conductivity of porous Media on D_i respectively. V be the the concentration of solute in the fluid. In addition we also require that U, V, (K(x,U)U_x-1) and DθV_x+V(KU_x-K) are continu ous at x=1. We prove the exisence, uniqueness and large time behavior of the problem by the method of reg ularization.
文摘In this paper,we consider the degenerate diffusion problem where Ω R^N is an open bounded domain with lipschta boundary Ω9. ψ(u) ∈ c^2[0,∞) ,ψ(s)>O,ψ′(s) >0, ψ′(s)≥0 ,whens>0;ψ(0)= 0,ψ′(0))≥ 0. u_0(x)∈ H_0~1(Ω) ∩ C^o(),u_o(x)≥0.g(s) is locally lipschitz continuous on [0,∞), g(0)=0. There exist a constant K,K> maxu_o(x), such that g(K) <0. |g(s)|/ψ(s)≤ M_o when 0≤s≤K ,where M_0 is a constant. We prove the existence and localization phenomena of weak solution of above problem. Under some additional conditions,we prove th uniqueness,contiouous and asymptotic behavier of weak solution.
