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We study the large time behavior of the solutions of the Cauchy problem for a semilinear heat equation,whereAssume that u is a solution of (P) satisfyingfor some constants C > 0 and A > 1. Then it is well known that the solution u behaves like the heat kernel. In this paper we give the ([K ] + 2)th order asymptotic expansion of the solution u, and reveal the relationship between the asymptotic profile of the solution u and the nonlinear term F. Here [K ] is the integer satisfying K − 1 < [K ] ≤ K .

This paper is concerned with a nonlinear integral equation) is a generalization of the heat kernel. We are interested in the asymptotic expansions of the solution of (P ) behaving like a multiple of the integral kernel G as t → ∞.

We study the existence and the large time behavior of global-intime solutions of a nonlinear integral equation with a generalized heat kernelwhere ϕ ∈ W ,∞ (R N ) and ∈ {0, 1, . . . }. The arguments of this paper are applicable to the Cauchy problem for various nonlinear parabolic equations such as fractional semilinear parabolic equations, higher order semilinear parabolic equations and viscous Hamilton-Jacobi equations.2010 Mathematics Subject Classification. Primary: 35A01, 35K55; Secondary: 35K91.

We consider a nonlinear parabolic equation with an exponential nonlinearity which is critical with respect to the growth of the nonlinearity and the regularity of the initial data. After showing the equivalence of the notions of weak and mild solutions, we derive decay estimates and the asymptotic behavior for small global-in-time solutions.

Let u be a solution of the Cauchy problem for the nonlinear parabolic equationand assume that the solution u behaves like the Gauss kernel as t → ∞. In this paper, under suitable assumptions of the reaction term F and the initial function ϕ, we establish the method of obtaining higher order asymptotic expansions of the solution u as t → ∞.

In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including $$ \left\{\begin{array}{ll} \partial_t u=\Delta u+v^p,\qquad & x\in\Omega,\,\,\,t>0,\\ \partial_t v=\Delta v+u^q, & x\in\Omega,\,\,\,t>0,\\ u=v=0, & x\in\partial\Omega,\,\,\,t>0,\\ (u(x,0), v(x,0))=(u_0(x),v_0(x)), & x\in\Omega, \end{array} \right. $$ where $p\ge 0$, $q\ge 0$, $\Omega$ is a (possibly unbounded) smooth domain in ${\bf R}^N$ and both $u_0$ and $v_0$ are nonnegative and locally integrable functions in $\Omega$. The supersolutions enable us to obtain optimal sufficient conditions for the existence of the solutions and optimal lower estimates of blow-up rate of the solutions

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