Solutions to a nonlinear heat equation with critical exponent

In this study we consider the nonlinear heat equation u_t = Δu + |u| p−1u on $W\times R+$ with some boundary conditions and the initial condition u (x, 0) = u₀( x) $\in$ L1($W$), where $W\subset Rn$ and p > 1. For the one dimensional case, it is well known for p < 3 that this problem has a local solution for any initial condition u₀ $\in$ L1($W$). But the existence and uniqueness of a local solution in L1 for the critical exponent p = 3 was widely open and this work is to answer this open question. First, we prove for the Cauchy problem that there is no local solution in L 1 for some u₀ $\in$ L1. Then using the nonlocal existence of the Cauchy problem by a cutoff function argument, we prove the nonlocal existence of a solution for the Dirichlet, problem which answers this open question. Moreover we generalize the nonlocal existence result for the n-dimensional case with the critical exponent $p=1+2n$. More general nonlinearity is also considered for Dirichlet boundary value problems. And also we prove the same result for the mixed boundary condition with the same initial data u₀. Finally, we establish the global existence in L1+&epsis; with || u₀||_1+&epsis; sufficiently small and &epsis; > 0, for the critical exponent p = 3 and n = 1. By using some numerical methods we also give some numerical simulations to these results.