Cauchy problem for a semilinear Éidel’man parabolic equation - Korkuna, O. E. (Ukrains’kyi Matematychnyi Zhurnal, 2008)
We obtain conditions for the existence and uniqueness of a generalized solution of the Cauchy problem for the equation$$u_1 + \sum_{|α|=|β|=2}(-1)^{|α|}D^{α}_x(a_{α β}(z, t)D_x^{β}u) - \sum_{|α|=|β|=1}(-1)^{|α|}D^{α}_y(b_{α β}(z, t)D_y^{β}u) +$$ $$+ \sum_{|α|=1}c_{α}(z, t) D^{α}_zu + c(z, t, u) = \sum_{|α|≤2}(-1)^{|α|}D^{α}_x f_{α}(z, t) - \sum_{|α|=1}D^{α}_y g_{α}(z, t)$$in Tikhonov's class.