Publication details

Double-phase parabolic equations with variable growth and nonlinear sources



Year of publication 2023
Type Article in Periodical
Magazine / Source Advances in Nonlinear Analysis
MU Faculty or unit

Faculty of Science

Keywords singular and degenerate parabolic equation; double phase problem; variable nonlinearity; nonlinear source
Description We study the homogeneous Dirichlet problem for the parabolic equations u(t) - div(A(z, vertical bar del u vertical bar)del u) = F(z, u, del u), z = (x, t) is an element of Omega x (0, T), with the double phase flux A(z, vertical bar del u vertical bar)del u (vertical bar del u vertical bar(p(z)-2) + a(z)vertical bar del u vertical bar(q(z) -2))del u and the nonlinear source F. The initial function belongs to a Musielak-Orlicz space defined by the flux. The functions a, p, and q are Lipschitz-continuous, a(z) is nonnegative, and may vanish on a set of nonzero measure. The exponents p, and q satisfy the balance conditions 2N/N+2 < p(-) <= p(z) <= q(z) < p(z) + r*/2 with r* = r* (p(-), N) p(-) = min((Q) over barT) p(z). It is shown that under suitable conditions on the growth of F(z, u, del u) with respect to the second and third arguments, the problem has a solution u with the following properties: u(t) is an element of L-2(Q(T)), vertical bar del u vertical bar(p(z)+delta) is an element of L-1(Q(T)) for every 0 <= delta < r*, vertical bar del u vertical bar(s(z)), a(z)vertical bar del u vertical bar(q(z)) is an element of L-infinity(0, T; L-1(Omega)) with s(z) = max{2, p(z)}. Uniqueness is proven under stronger assumptions on the source F. The same results are established for the equations with the regularized flux A(z, (epsilon(2) + vertical bar del u vertical bar(2))(1/2))del u, epsilon > 0.
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