ZOOM Link: https://snu-ac-kr.zoom.us/j/99718630591?pwd=MitkTmNxOUl3TytJZW5NR2FtSXFVZz09
ID: 997 1863 0591
We consider a parabolic PDE with Dirichlet boundary condition and monotone operator $A$ with non-standard growth controlled by an $N$-function depending on time and spatial variable. We do not assume continuity in time for the $N$-function. Using an additional regularization effect coming from the equation, we establish the existence of weak solutions and in the particular case of isotropic $N$-function, we also prove their uniqueness. This result is based on the approximation theory in Bochner-Orlicz-Museliak-Sobolev spaces. Due to the structure of the problem, we obtain and additional regularization in time variable directly from the equations, which allows us to consider rather general-in-time $N$-function. This general result applies to equations studied in the literature like $p(t,x)$-Laplacian and double-phase problems. Moreover, we show, how the developed approximation method can be used for the elliptic case to avoid the so-called Lavrientev phenomenon.