Parabolic Equations in Orlicz Spaces

Abstract

An approximation theorem in inhomogeneous Orlicz–Sobolev spaces is proved which allows a second‐order parabolic equation in Orlicz spaces to be solved. A trace result is also given which shows that the solutions are continuous with respect to time.

Strongly nonlinear parabolic problems in Musielak-Orlicz-Sobolev spaces

“…The solvability of (1) in this setting is proved by Donaldson [12] for g ≡ 0 and by Robert [23] for g ≡ g(x, t, u) when A is monotone, t 2 ≪ M (t) and M satisfies a ∆ 2 condition and also by Elmahi [14] for g = g(x, t, u, ∇u) when M satisfies a ∆ as application of some L M compactness results in W 1,x L M (Q), see [13]. The solvability of (1) in this setting is proved by Elmahi-Meskine [16] for g ≡ 0 and for g ≡ g(x, t, u, ∇u) in [15], without assuming any restriction on the Nfunction M .…”

“…Go back to approximate equations (16) and use w j χ (0,τ ) , for every τ ∈ [0, T ](which belongs to W 1,x 0 L ϕ (Q)) as a test function one has ∂u n ∂t , w j Qτ + Qτ a(x, t, u n , ∇u n )∇w j dxdt + Qτ g n (x, t, u n , ∇u n )w j dxdt = f, w j Qτ , which implies that …”

“…For some classical and recent results on elliptic and parabolic problems in Orliczsobolev spaces and a Musielak-Orlicz-Sobolev spaces, we refer to [1,2,3,6,12,14,15,16,24].…”

Nonlinear parabolic problems in Musielak–Orlicz spaces

“…This will be a crucial fact in the existence proof, in particular showing the energy equality, which is necessary for the limit passage in nonlinear term. This kind of approximation theorem in case of classical Orlicz spaces was proved in [13]. Before formulating the theorem let us define the space V M as follows…”

“…[4,13,14]. All of them concern the case of classical spaces, namely Orlicz spaces with an N−function dependent only on |ξ| without the dependence on (t, x).…”