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The function $f(x):=u(x,t)-cx/t$ decreases, thus it$u(x,t)=f(x)+cx/t$ is locallya sum of two monotone functions, so belongs to local BV.
The function $f(x):=u(x,t)-cx/t$ decreases, thus it is locally BV.
The function $f(x):=u(x,t)-cx/t$ decreases, thus $u(x,t)=f(x)+cx/t$ is a sum of two monotone functions, so belongs to local BV.
