Asymptotic behavior of a quasilinear parabolic equation in a band domain with unbounded boundary slopes

In this paper we consider a quasilinear parabolic equation in a band domain with Robin boundary conditions. The equation arises from the heat equation and some curvature flows. We will show that the equation has three types of translating solutions according to the form of the equation: the first type has a <italic>power-function-like</italic> profile, the second type has a <italic>Grim Reaper-like</italic> profile, and the third type has a <italic>cup-like</italic> profile. We then show that the solution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u"> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding="application/x-tex">u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the initial-boundary value problem tends to infinite as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">t\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis x comma t right-parenthesis minus u left-parenthesis 0 comma t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> − </mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u(x,t)-u(0,t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> tends to vertical line in the first type case, and tends to the corresponding translating solutions in the second and third type cases.