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Periodic solutions of the first boundary value problem for a linear and weakly nonlinear heat equation

Věnceslava ŠťastnováOtto Vejvoda — 1968

Aplikace matematiky

One investigates the existence of an ω -periodic solution of the problem u t = u x x + c u + g ( t , x ) + ϵ f ( t , x , u , u x , ϵ ) , u ( t , 0 ) = h 0 ( t ) + ϵ χ 0 ( t , u ( t , 0 ) , u ( t , π ) ) , u ( t , π ) = h 1 ( t ) + ϵ χ 1 ( t , u ( t , 0 ) , u ( t , π ) ) , provided the functions g , f , h 0 , h 1 , χ 0 , χ 1 are sufficiently smooth and ω -periodic in t . If c k 2 , k natural, such a solution always exists for sufficiently small ϵ > 0 . On the other hand, if c = l 2 , l natural, some additional conditions have to be satisfied.

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