The decay of the solutions for the heat equation with a potential

Higher Order Asymptotic Expansion for the Heat Equation with a Nonlinear Boundary Condition

“…Furthermore, combining (1.5), and (2.1)-(2.3) we have uðx; tÞ ¼ ðSðtÞjÞðxÞ þ 2k The end of this subsection we introduce some operators P K ðtÞ and Q K ðtÞ. Following [6] and [9], for any k b 0 and t > 0, we introduce a linear operator P k ðtÞ on L for all t > 0 (see [6]). This operators are key of our proof, in particular (2.8) and (2.9) are crucial properties in our analysis.…”

“…Furthermore, as far as we know, there are no results treating higher order asymptotic expansions of the solution of (1.1) even if j A C with k A R and p > 1 þ 2=N, in [9], improving the argument in [6], the author of this paper and Ishige established the useful method which derives higher order asymptotic expansions of the solutions for the Cauchy problem (1.10). The argument in [9] is applicable to the large class of the nonlinear parabolic equations in the whole space (see [8]).…”

“…Assertion (1.4) gives the large time behavior, in particular the first term of asymptotic expansions, of the solution u of (1.1) satisfying (1.3). The large time behavior of the solutions for nonlinear parabolic problem including the problem (1.1) is influenced by the behavior of the initial datum at the spatial infinity and the nonlinearity, and has been studied in many papers (see [6]- [11], [13], and references therein). In fact, for the problem with p > 1 þ 1=N, in [11], applying the entropy dissipation method, the author of this paper proved that, under condition (1.2) with K ¼ 2, if the solution u of (1.6) satisfies (1.3), then, for any q A ½1; y, Furthermore, in [7], the author of this paper and Ishige showed that, under…”

Refined asymptotic profiles for a semilinear heat equation

“…The limit (1.5) gives the first term of the asymptotic expansion of the solution u of (1.1). For the Cauchy problem (1.1), the authors of this paper and Ishiwata in [6] studied the decay rate of L q (R N )-norm of the remainder term R(x, t) := u(x, t) − MG(x, 1 + t), (1.6) and discussed the relationship between the decay rate of R(t) L q (R N ) and the exponent K in condition (1.2). Furthermore they applied their arguments to the Cauchy problem for the semilinear heat equation,…”

“…(see for example [6]). The limit (1.5) gives the first term of the asymptotic expansion of the solution u of (1.1).…”