Construction of fundamental solutions of an abstract nonlinear parabolic equation in a neighborhood of a bifurcation point

A nonlinear equation of parabolic type with functions taking values in a Banach space is studied. A family of solutions called a fundamental family is constructed in a neighborhood of a bifurcation point. It is shown that as $t\to\infty$ the fundamental solutions tend either to zero or to some steady-state solution of the nonlinear equation. Conditions are investigated under which the solutions of Cauchy problems behave like fundamental solutions.
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