Global existence and uniqueness of the solution to a nonlinear parabolic equation

Abstract

Consider the equation

 u’ (t)  -  u + | u |^p u = 0, u(0) = u₀(x), (1),

where u’ := du/dt , p = const > 0, x E R³, t > 0.

 Assume that u₀ is a smooth and decaying function,

          ||u₀|| =            sup             |u(x, t)|.

                                        x E R₃ ,t E R+     

It is proved that problem (1) has a unique global solution and this

solution satisfies the following estimate

                             ||u(x, t)|| < c,

where c > 0 does not depend on x, t.