When it attempted to evaluate
D[u[0,t],x]==0
there was no x to differentiate with respect to and thus the result was zero and thus that result was immediately True. Likewise for two more of your boundary conditions.
This tries to correct those issues.
DSolve[{1/a*D[u[x,t],t]+tq/a*D[u[x,t],t,t]==D[u[x,t],x,x]+tt*D[u[x,t],x,x,t]+
1/lamda*(5977113*(HeavisideTheta[x-L/2+R]-HeavisideTheta[x-L/2-R])),
(D[u[x,t],x]/.x->0)==(D[u[x,t],x]/.x->L)==0,D[u[x,t0],x]==0,u[x,t0]==T0},u[x,t],{x,t}]
Please check that very carefully to make certain that it is correct.
When Mathematica echos back your input that indicates that it cannot solve a problem. Unfortunately this seems to indicate that Mathematica can find a solution to your problem.
You might consider whether you could assign numeric constants to all your parameters and do a numeric NDSolve.
