*Mathematical Methods in the Applied Sciences*New Blow-up Conditions to $p$-Laplace Type Nonlinear Parabolic
Equations under Nonlinear Boundary Conditions

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 \(\Gamma_{2}\times(0,t^{*})\) such that \(\Gamma_{1}\cup\Gamma_{2}=\partial\Omega\), where \(f\) and \(h\) are real-valued \(C^{1}\)-functions. To discuss blow-up solutions, we introduce new conditions:

For each \(x\in\Omega\), \(z\in\partial\Omega\), \(t>0\), \(u>0\), and \(v>0\),
for some constants \(\alpha\), \(\beta_{1}\), \(\beta_{2}\), \(\gamma_{1}\), \(\gamma_{2}\), and \(\delta\) satisfying
where \(\rho_{m}:=\inf_{w>0}\rho(w)\), \(P(v)=\int_{0}^{v}\rho(w)dw\), \(F(x,t,u)=\int_{0}^{u}f(x,t,w)dw\), and \(H(x,t,u)=\int_{0}^{u}h(x,t,w)dw\). Here, \(\lambda_{R}\) is the first Robin eigenvalue and \(\lambda_{S}\) is the first Steklov eigenvalue for the \(p\)-Laplace operator, respectively.

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