Hölder estimates of solutions of biological population equations
The author considers the Cauchy problem \[ \begin{aligned} & u_t=G(u)_{xx}+G(u)_yy +f(t,x,y,u),\\ & u(x,y,0)=u_0(x,y),\end{aligned} \] which arise in the spread of biological populations. The main result is the following: if \(G'(u)>0\), \(G(0)=0\), \(0\leq u_0(x,y)\leq M\), \(f\in C^1(\mathbb{R}^+\times \mathbb{R}^2\times \mathbb{R}^+)\) and \(f=0\) at \(u=0\), then there exists a weak solution. If \(G(u)=u^2\) and \(f_x\), \(f_y\), \(f_u\), \(f/u\) satisfy a boundedness condition, then this solution is Hölder continuous: with the exponents 2/3 with respect to \(x,y\) and 1/4 with respect to \(t\).
scientific article; zbMATH DE number 1531187
scientific article