On principal eigenvalues for periodic parabolic problems with optimal condition on the weight function (Q5952263)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\), and let \(m(x,t)\) be a \(T\)-periodic function such that its restriction to \(\Omega\times (0,T)\) belongs to \(L^s((0,T), L^v(\Omega))\) for some \(v>N/2\) and \(s>2v(2v-N)^{-1}\). The Dirichlet periodic parabolic eigenvalue problem with the weight function \(m(x,t)\) NEWLINE\[NEWLINE\begin{cases} u_t-\text{div} (A\nabla u)+\langle b,\nabla u\rangle+ c_0u= \lambda mu\text{ in }\Omega \times \mathbb{R},\\ u=0 \text{ on }\Omega\times\mathbb{R},\;u(x,t)= u(x,t+T)\text{ a.e. }(x,t)\in\Omega \times\mathbb{R},\end{cases} \tag{1}NEWLINE\]NEWLINE is considered; \(A=(a_{ij} (x,t))\) is a \(N\times N\) matrix, \(a_{ij}(x,t)\) being some \(T\)-periodic functions in \(t\), \(a_{ij}= a_{ji}\), \(b=(b_1(x,t), \dots,b_N(x,t))\), \(b_j(x,t)\) being also \(T\)-periodic functions, \(c_0\) is a nonnegative function, \(\langle , \rangle\) denotes the standard inner product on \(\mathbb{R}^N\). In connection with \(a_{ij}\), \(b_j\) and \(c_0\) it is assumed that these functions belong to certain functional spaces and \(\sum_{i,j} a_{ij}(x,t) \xi_i\xi_j \geq |\xi|^2\), for some positive constant \(\alpha_0\) and all \((x,t)\in \Omega\times \mathbb{R}\), \(\xi=(\xi_1, \dots,\xi_N) \in\mathbb{R}^N\).NEWLINENEWLINENEWLINEThe main result of this paper is a necessary and sufficient condition on \(m\) for the existence of a principal positive eigenvalue for (1); this eigenvalue is unique and algebraically simple.