On the Cauchy problem for a second order semilinear parabolic equation with factored linear part (Q1194434)

The following Cauchy problem of factored type is studied in a Banach space \({\mathcal X}\) \[ \left({d\over dt}-A_ 1\right) \left({d\over dt}-A_ 0\right) u(t)=f\bigl(t,u(t),u'(t)\bigr),\quad u(0)=u_ 0,\;u'(0)=u_ 1.\tag{1} \] In (1) \(A_ 0\), \(A_ 1\) are infinitesimal generators of analytic semigroups in \({\mathcal X}\) and the function \(f\) and the initial conditions \(u_ 0\), \(u_ 1\) satisfy suitable regularity assumptions. It is shown that (1) has a strict solution \(u=u(t)\) which is also Hölder continuous. Some examples and applications to these results on second order equations that can be reduced to equations of factored type are given.