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On a Fokker-Planck equation arising in population dynamics. (English) Zbl 0921.45007

The existence and uniqueness of the solution of the nonlinear initial-boundary value problem

t u= x {M(u;t,x)u+ x (D(u;t,x)u)}in(0,T)×(0,1)u t=0 =u 0 M(u)u+ x (D(u)u)=0onx{0,1}

is established in L q (0,T;W 1/q (0,1)) for all 1q<4 3 and nonnegative initial data u 0 L 1 (0,1). This problem models the evolution of certain properties in populations of social organisms. Departing from well-known properties of L 2 -solutions of the corresponding linearized problem, the Riesz-Schauder fixed point theorem is applied to solve the above nonlinear problem for nonnegative L 2 initial data u 0 . Deriving some estimates depending only on the L 1 norm of u 0 and applying approximation by nonnegative initial data in C 0 (0,1) as well as a compactness lemma, the existence result is extended to arbitrary nonnegative L 1 initial data.

MSC:
45K05Integro-partial differential equations
92D25Population dynamics (general)
45G10Nonsingular nonlinear integral equations
45M20Positive solutions of integral equations