Hello!!!

I want to show that if $v_i(t,x_i), i=1, \dots, n$ are solutions of the equations $v_{i x_i x_i} -v_{it}=0, i=1, \dots,n$ respectively, then $v=v_1 v_2 \cdots v_n(v=v(t,x))$ is a solution of $\Delta v-v_t=0$.

That's what I have done so far.

$$v_x=(v_{1x} v_2 \cdots v_n)+(v_1 v_{2x} \cdots v_n)+ \dots+ (v_1 v_2 \cdots v_{nx})$$

$$v_{xx}=(v_{1xx} v_2 \cdots v_n+ v_{1x} v_{2x} \cdots v_{n}+ \dots+ v_{1x} v_2 \cdots v_{nx})+ \dots+ (v_{1x} v_2 \cdots v_{nx}+ \dots+ v_1 v_2 \cdots v_{nxx} )$$

So do we have to include now at $v_{xx}$ the sum $\sum_{i=1}^n v_{i x_i x_i}$?

How can we use the fact that $v_{i x_i x_i} -v_{it}=0, i=1, \dots,n$ ?