Global well-posedness and exponential decay rates for a KdV–Burgers equation with indefinite damping

We consider the KdV–Burgers equation ut+uxxx−uxx+λu+uux=0ut+uxxx−uxx+λu+uux=0 and its linearized version ut+uxxx−uxx+λu=0ut+uxxx−uxx+λu=0 on the whole real line. We investigate their well-posedness their exponential stability when λ is an indefinite damping.