We are interested in existence and uniqueness of renormalized
solutions to the nonlinear initial-boundary value problem
\begin{align*}
\partial_t \beta(x,u) - \operatorname{div} (a(x,Du) + F(u)) = f & \text{ in } Q_T= (0,T) \times \Omega\\
u=0 & \text{ on } \Sigma_T= (0,T) \times \partial \Omega \\
\beta( \cdot,u(0, \cdot) ) = b_0 & \text{ on } \Omega,
\end{align*}
where
- \(\beta\colon \Omega \times \mathbb{R} \rightarrow \mathbb{R}\) is
a monotone single-valued Carathéodory function,
- \(F\colon \mathbb{R} \rightarrow \mathbb{R}^N\) is a locally
Lipschitz continuous function,
- \(b_0\) and \(f\) are integrable given data, and
- the Carathéodory vector field \(a\colon \Omega \times
\mathbb{R}^N \rightarrow \mathbb{R}^N\) is monotone w.r.t. \(\xi
\in \mathbb{R}^N\) and satisfies generalized growth and
coerciveness conditions of the form \( a(x, \xi) \cdot \xi \geq
c_a (M(x,\xi) + M^*(x, a(x, \xi)) - a_0(x) \quad \text{a.e. } x
\in \Omega, \forall \xi \in \mathbb{R}^N\),
with \(c_a >0\), \(a_0 \in L^1(\Omega)\), \(M: \Omega \times
\mathbb{R}^N \rightarrow \mathbb{R}\) being a generalized
\({\mathcal{N}}\)-function with complementary function \(M^*\). Our
setting includes parabolic problems involving the \(p(x)\)- and also
the anisotropic \(p=(p_1, \ldots, p_N)\)-Laplacian with variable
exponents essentially larger than 1.
The appropriate functional setting involves generalized
Musielak-Orlicz spaces \(L_M(\Omega;\mathbb{R}^N)\) which, in
general, are neither separable nor reflexive. Therefore, classical
monotonicity and truncation techniques have to be appropriately
adapted to the non-reflexive and non-separable functional setting.
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