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On the blow-up set for non-Newtonian equation with a nonlinear boundary condition. (English) Zbl 1207.35247

Summary: We identify the blow-up set of solutions to the problem \(u_t=(|u_x|^{p-2}u_x)_x\), \(x>0\), \(t\in(0,T)\), \(-|u_x|^{p-2}u_x(0,t)= u^{p-1}(0,t)\), \(t\in (0,T)\), and \(u(x,0)= u_0(x)\geq 0\), \(x>0\), where \(p>2\). We obtain that the blow up set \(B(u)\) satisfies \([0,p(p-1)/(p-2))\subset B(u)\subset [0,p(p-1)/(p-2)]\). The proof is based on the analysis of the asymptotic behavior of self-similar representation and on the comparison methods.

MSC:

35Q35 PDEs in connection with fluid mechanics
76R50 Diffusion
76A05 Non-Newtonian fluids
35B44 Blow-up in context of PDEs
35D30 Weak solutions to PDEs
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References:

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