The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations u t = Δ u + ψ ( t ) f ( u ) , in Ω × ( 0 , t ∗ ) , under the Dirichlet boundary condition on a bounded domain. In fact, this has remained as an open problem for a few decades, even for the case f ( u ) = u p . As a matter of fact, we prove: there is no global solution for any initial data if and only if ∫ 0 ∞ ψ ( t ) f ( ‖ S ( t ) u 0 ‖ ∞ ) ‖ S ( t ) u 0 ‖ ∞ dt = ∞ for every nonnegative nontrivial initial data u 0 ∈ C 0 ( Ω ) . Here, ( S ( t ) ) t ≥ 0 is the heat semigroup with the Dirichlet boundary condition.

A necessary-sufficient condition for the existence or nonexistence of global solutions to the following reaction-diffusion equations { u t = Δ u + ψ ( t ) u p , in R N ×( 0 , t ∗ ) , u ( ⋅ , 0 )= u 0 ≥ 0 , in R N , has not been known and remained as an open problem for a few decades. The purpose of this paper is to resolve this problem completely, even for more general source ψ( t) f( u) as follows: There is a global solution to the equation if and only if ∫ 0 ∞ ψ ( t ) f ( ‖ S ( t ) u 0 ‖ ∞ ) ‖ S ( t ) u 0 ‖ ∞ dt < ∞ for some nonnegative and nontrivial u 0 ∈ C 0 ( R N ) . Here, ( S ( t ) ) t ≥ 0 is the heat semigroup on R N .

The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations \[ u_{t}=\Delta u+\psi(t)f(u),\,\,\mbox{ in }\Omega\times (0,t^{*}), \] under the Dirichlet boundary condition on a bounded domain. In fact, this has remained as an open problem for a few decades, even for the case $f(u)=u^{p}$. As a matter of fact, we prove:\\ \[ \begin{aligned} &\mbox{there is no global solution for any initial data if and only if }\\ &\mbox{the function } f \mbox{ satisfies}\\ &\hspace{20mm}\int_{0}^{\infty}\psi(t)\frac{f\left(\epsilon \,\left\| S(t)u_{0}\right\|_{\infty}\right)}{\left\| S(t)u_{0}\right\|_{\infty}}dt=\infty\\ &\mbox{for every }\,\epsilon>0\,\mbox{ and nonnegative nontrivial initial data }\,u_{0}\in C_{0}(\Omega). \end{aligned} \] Here, $(S(t))_{t\geq 0}$ is the heat semigroup with the Dirichlet boundary condition.

In this paper, we study blow-up phenomena of the following p-Laplace type nonlinear parabolic equations under nonlinear mixed boundary conditions and u = 0 on Γ₂ × (0, t*) such that Γ₁ ∪ Γ₂ = ∂Ω, where f and h are real-valued C¹-functions. To discuss blow-up solutions, we introduce new conditions: For each x ∈ Ω, z ∈ ∂Ω, t > 0, u > 0, and v > 0, for some constants α, β₁, β₂, γ₁, γ₂, and δ satisfying where ρm := infw > 0ρ(w), P(v)=∫₀vρ(w)dw, F(x, t, u)=∫₀uf(x, t, w)dw, and H(x, t, u)=∫₀uh(x, t, w)dw. Here, λR is the first Robin eigenvalue and λS is the first Steklov eigenvalue for the p-Laplace operator, respectively.