Abstract
An existence result of a renormalized solution for a class of non- linear parabolic equations in Orlicz spaces is proved. No growth assumption is made on the nonlinearities.
Highlights
In this paper we consider the following problem: (1.1) ∂b(x, ∂t u) −div a(x, t, u, ∇u) + Φ(u)=f in Ω × (0, T ), (1.2)b(x, u)(t = 0) = b(x, u0) in Ω, (1.3)u = 0 on ∂Ω × (0, T ), where Ω is a bounded open subset of RN and T > 0, Q = Ω × (0, T )
Let b be a Caratheodory function (see assumptions (3.1)-(3.2) of Section 3), the data f and b(x, u0) in L1(Q) and L1(Ω) respectively, Au = −div a(x, t, u, ∇u) is a Leray-Lions grows olipkeeraMto−r1dMefi(nβeK4d|∇onu|W) w01,ixtLhMre(sΩp)e,ctMtois∇aun, appropriate N -function and which but which is not restricted by any growth condition with respect to u (see assumptions (3.3)-(3.6))
Redwane [34, 35] in the case where Au = −div a(x, t, u, ∇u) is a Leray-Lions operator defined on Lp(0, T ; W01,p(Ω)), the existence of renormalized solution in Orlicz spaces has been proved in E
Summary
In this paper we consider the following problem:. u = 0 on ∂Ω × (0, T ), where Ω is a bounded open subset of RN and T > 0, Q = Ω × (0, T ). The function Φ is just assumed to be continuous on R Under these assumptions, the above problem does not admit, in general, a weak solution since the fields a(x, t, u, ∇u) and Φ(u) do not belong in (L1loc(Q)N in general. Redwane [34, 35] in the case where Au = −div a(x, t, u, ∇u) is a Leray-Lions operator defined on Lp(0, T ; W01,p(Ω)), the existence of renormalized solution in Orlicz spaces has been proved in E. Note that here we extend the results in [34, 32] in three different directions: we assume b(x, u) depend on x , and the growth of a(x, t, u, ∇u) is not controlled with respect to u and we prove the existence in Orlicz spaces.
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