In this paper, we prove the existence and the uniqueness of renormalized solution to a nonlinear multivalued parabolic problem β(u)t - div a(x,∇ u) ∋ f , with homogeneous Dirichlet boundary conditions and L1-data. The functional setting involves Lebesgue and Sobolev spaces with variable exponent. Some a-priori estimates are used to obtain our results.
We prove two different kind of results for the magnetic elliptic operator which are the spectral gap of the magnetic elliptic functional energy and the inverse Poincare inequality which is used to prove that if u vanish on a set of positive measure E then, u has a zero of infinite order at almost every point of E.
In this work, we study a class of abstract non-autonomous partial functional differential equations with infinite delay. Our main results concern the local existence of the mild solution which can blow up at the finite time. The unbounded operators associated to the non-autonomous system are assumed to be stable family which generates C
0-semigroups while the nonlinear part is supposed to be continuous. Under Lipschitz condition on the nonlinear term of the equation, we prove the existence and uniqueness of the mild solution. For illustration, we provide an example for some reaction-diffusion non-autonomous partial functional differential equations involving infinite delay.
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