(untitled) (European Journal of Pure and Applied Mathematics, 2024)
We use recent approximation results in BV space to derive L^∞ bounds for u, L^∞ bounds in time for the BV seminorm |Du| of u, and L^2 bounds for u_t for the weak solution uC([0,∞ );L^2( Ω ) BV( Ω ) ), % Ω %TCIMACRO211d % %BeginExpansion R %EndExpansion ^N open and bounded, of the time flow equation* ut=div abla _pφ (x,Du)-λ (u-u_0), λ >0, u(0,x)=u_0. equation*% We assume Neumann boundary condition and φ (x,p) is in a class of linear growth functions in p. Importantly, φ (· ,p)L^1( Ω ) in contrast to the classical results stated in ACM where φ has a continuity assumption in the x variable. We also use the convergence of the solution above to derive an L^∞ bound for the solution u^* to the corresponding stationary problem, since u(t)→ u^* in L^1( Ω ) .