Existence Results For A Semilinear Parabolic Equation

In this paper, we study the global existence of solution of the following semilinear parabolic Cauchy problemu_t+ sum form i=1 to n (?)f_i(u)/(?)x_i+f₀(u) =â–³u, in (0, +∞)×Rⁿwith the initial conditionu(x, 0) =φ(x), on Rⁿin L^q(Rⁿ), where f_i(u)∈C¹, i = 0,1,..., n.We make the following assumptions:|f_i(u)|≤C_i|u|^β, |f′_i(u)|≤C_i|u|^β-1, i=1,...,n|f₀(u)|≤C₀|u|γ, |f′₀(u)|≤C₀|u|^γ-1.where C_i,C₀ are some positive constants andβ,γ> 1 are fixed parameters. Assume q = n(γ-1)/2>1, and that given anyφbe in L^q(Rⁿ), where‖φ‖_L^q is sufficiently small, then there exists a global solution in L^q .