Abstract
We prove an existence result for solutions of nonlinear parabolic inequalities with data in Orlicz spaces.
Highlights
Let Ω be an open bounded subset of RN, N ≥ 2, let Q be the cylinder Ω × (0, T) with some given T > 0
Where A(u)=−div(a(x, t, u, ∇u)) is a Leray-Lions operator defined on D(A) ⊂ W01,xLM(Ω), with M is an N-function, and χ is a given data
In the variational case (i.e., where χ ∈ W−1,xEM(Ω)), it is well known that the solvability of (1.1) is done by Donaldson [2] and Robert [11] when the operator A is monotone, t2 M(t), and M satisfies a Δ2 condition, and by the recent work [3] for the general case
Summary
Let Ω be an open bounded subset of RN , N ≥ 2, let Q be the cylinder Ω × (0, T) with some given T > 0. In the L1 case, an existence theorem is given in [4]. The techniques used in [4] do not allow us to adapt it for parabolic inequalities. It is our purpose in this paper to solve the obstacle problem associated to (1.1) in the case where χ ∈ L1(Q) + W−1,xEM(Q) and without assuming any growth restriction on M. For some classical and recent results in the setting of Orlicz spaces dealing with elliptic and parabolic equations, the reader can be referred to [8, 10, 12,13,14]
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