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Zbl 0921.45007
On a Fokker-Planck equation arising in population dynamics.
(English)
[J] Rev. Mat. Complut. 11, No.2, 353-372 (1998). ISSN 1139-1138; ISSN 1988-2807/e

The existence and uniqueness of the solution of the nonlinear initial-boundary value problem $$\cases\partial_tu=\partial_x\{M(u;t,x)u+\partial_x(D(u;t,x)u)\}\text{ in }(0,T)\times(0,1)\\ \left.u\right|_{t=0}=u_0\\ M(u)u+\partial_x(D(u)u)=0\text{ on }x\in\{0,1\}\endcases$$ is established in $L^q(0,T;W^{1/q}(0,1))$ for all $1\le q<{4\over 3}$ and nonnegative initial data $u_0\in L^1(0,1)$. 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 $C_0^\infty(0,1)$ as well as a compactness lemma, the existence result is extended to arbitrary nonnegative $L^1$ initial data.
[Cornelis van der Mee (Cagliari)]
MSC 2000:
*45K05 Integro-partial differential equations
92D25 Population dynamics
45G10 Nonsingular nonlinear integral equations
45M20 Positive solutions of integral equations

Keywords: Fokker-Planck equation; population dynamics; positive solutions; nonlinear intitial-boundary value problem; compactness

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