Cauchy problem for a semilinear Éidel’man parabolic equation

Abstract

We obtain conditions for the existence and uniqueness of a generalized solution of the Cauchy problem for the equation $$u_1 + \sum_{|\alpha|=|\beta|=2}(-1)^{|\alpha|}D^{\alpha}_x(a_{\alpha \beta}(z, t)D_x^{\beta}u) - \sum_{|\alpha|=|\beta|=1}(-1)^{|\alpha|}D^{\alpha}_y(b_{\alpha \beta}(z, t)D_y^{\beta}u) +$$ $$+ \sum_{|\alpha|=1}c_{\alpha}(z, t) D^{\alpha}_zu + c(z, t, u) = \sum_{|\alpha|\leq2}(-1)^{|\alpha|}D^{\alpha}_x f_{\alpha}(z, t) - \sum_{|\alpha|=1}D^{\alpha}_y g_{\alpha}(z, t)$$ in Tikhonov's class.