On the game controlled by the generalized heat conduction equation (Q2742929)

Two players with opposite objectives choose the initial function \(z(0,x)=f(x)\) of the process NEWLINE\[NEWLINEz_t=P(x)z_{xx} +Q(x)z_x+R(x)z,\quad t\in(0,T), x\in(a,b),NEWLINE\]NEWLINE where \(z(t,a)=z(t,b)=0\) and \(P(x)>0\), in the following way NEWLINE\[NEWLINEf(x)=\begin{cases} u(x), &x\in[a,c]\\v(x),& x\in[c,b]\end{cases}.NEWLINE\]NEWLINE The second player chooses the function \(v(x)\) and then the first player chooses the function \(u(x)\) so that \(f(x)\) to be a smooth function and \(z(T,x)\) to be equal to a given smooth function. It is proved that if \(c\in[a,b)\) then the winner is the player I, otherwise it is the player II.