The sharp lower bound of the lifespan of solutions to semilinear wave   equations with low powers in two space dimensions
  This paper is devoted to a proof of the conjecture in Takamura(2015) on the lower bound of the lifespan of solutions to semilinear wave equations in two space dimensions. The result is divided into two cases according to the total integral of the initial speed. 
Takuto Imai, Masakazu Kato, Hiroyuki Takamura, Kyouhei Wakasa
math.AP