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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

$\left\{\begin{array}{c}{\partial }_{t}u={\partial }_{x}\left\{M\left(u;t,x\right)u+{\partial }_{x}\left(D\left(u;t,x\right)u\right)\right\}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\left(0,T\right)×\left(0,1\right)\hfill \\ {u|}_{t=0}={u}_{0}\hfill \\ M\left(u\right)u+{\partial }_{x}\left(D\left(u\right)u\right)=0\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}x\in \left\{0,1\right\}\hfill \end{array}\right\$

is established in ${L}^{q}\left(0,T;{W}^{1/q}\left(0,1\right)\right)$ for all $1\le q<\frac{4}{3}$ and nonnegative initial data ${u}_{0}\in {L}^{1}\left(0,1\right)$. 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 ${C}_{0}^{\infty }\left(0,1\right)$ as well as a compactness lemma, the existence result is extended to arbitrary nonnegative ${L}^{1}$ initial data.

##### MSC:
 45K05 Integro-partial differential equations 92D25 Population dynamics (general) 45G10 Nonsingular nonlinear integral equations 45M20 Positive solutions of integral equations