Joint BCAM-UPV/EHU Analysis and PDE seminar: Test function method in KdV hierarchy

One can extend KdV equation from 1 dimension to higher dimension finding many equations that appears in mathematical physics. For example one has KP, BLMP, YTSF, CBS equations. They inherits from KdV a quasilinear evolution structure: L(∂t, ∇)u + Q(u, ∇u) = 0 with linear L and Q in divergence form.  We imagine to perturb one of these equations with a semilinear forcing term. We gain Lu + Q(u, ∇u) = N(u, ∇u) with polynomial growth for N. The interaction between L, Q, N destroy the symmetry of the equations and for suitable N one can prove non-existence of solutions and lifespan estimate. The test function method is the first idea for proving this, but due to the peculiar for of L and Q something new appears in initial data condition.