On a mixed problem for a linear coupled system with variable coefficients

We prove existence, uniqueness and exponential decay of solutions to the mixed problem $$u''(x,t)-mu(t)Delta u(x,t)+sum_{i=1}^n {partial hetaoverpartial x_i}(x,t)=0 $$ $$ heta'(x,t)-Delta heta(x,t) +sum_{i=1}^n {partial u'overpartial x_i}(x,t)=0,,$$ with a suitable boundary damping, and a positive real-valued function $mu$.