In this work we consider several semilinear damped wave equations with “subcritical” nonlinearities, focusing on studying lifespan estimates for energy solutions. Our main concern is on equations ...with scale-invariant damping and mass. By imposing different assumptions on the initial data, we prove lifespan estimates from above, distinguishing between “wave-like” and “heat-like” behaviours. Furthermore, we conjecture logarithmic improvements for the estimates on the “transition surfaces” separating the two behaviours. As a direct consequence, we reorganize the blow-up results and lifespan estimates for the massless case, and we obtain in particular improved lifespan estimates for the one dimensional case, compared to the known results.
We also study semilinear wave equations with scattering damping and negative mass term, finding that if the decay rate of the mass term equals to 2, the lifespan estimate coincides with the one in a special case of scale-invariant damped equation.
The main tool employed in the proof is a Kato's type lemma, established by iteration argument.
We study the Cauchy problem for a generalized derivative nonlinear Schrödinger equation with the Dirichlet boundary condition. We establish the local well-posedness results in the Sobolev spaces H1 ...and H2. Solutions are constructed as a limit of approximate solutions by a method independent of a compactness argument. We also discuss the global existence of solutions in the energy space H1.
We determine the critical blow-up exponent for a Keller–Segel-type chemotaxis model, where the chemotactic sensitivity equals some nonlinear function of the particle density. Assuming some growth ...conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution. Furthermore, we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails.
We consider the Yamabe equation
Δ
u
+
n
(
n
−
2
)
4
|
u
|
4
n
−
2
u
=
0
in
R
n
,
n
⩾
3
. Let
k
⩾
1
and
ξ
j
k
=
(
e
2
j
π
i
k
,
0
)
∈
R
n
=
C
×
R
n
−
2
. For all large
k we find a solution of the form
...u
k
(
x
)
=
U
(
x
)
−
∑
j
=
1
k
μ
k
−
n
−
2
2
U
×
(
μ
k
−
1
(
x
−
ξ
j
)
)
+
o
(
1
)
, where
U
(
x
)
=
(
2
1
+
|
x
|
2
)
n
−
2
2
,
μ
k
=
c
n
k
2
for
n
⩾
4
,
μ
k
=
c
k
2
(
log
k
)
2
for
n
=
3
and
o
(
1
)
→
0
uniformly as
k
→
+
∞
.
In this paper, we study a class of time-delayed reaction–diffusion equation with local nonlinearity for the birth rate. For all wavefronts with the speed
c
>
c
∗
, where
c
∗
>
0
is the critical wave ...speed, we prove that these wavefronts are asymptotically stable, when the initial perturbation around the traveling waves decays exponentially as
x
→
−
∞
, but the initial perturbation can be arbitrarily large in other locations. This essentially improves the stability results obtained by Mei, So, Li and Shen M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579–594 for the speed
c
>
2
D
m
(
ε
p
−
d
m
)
with small initial perturbation and by Lin and Mei C.-K. Lin, M. Mei, On travelling wavefronts of the Nicholson's blowflies equations with diffusion, submitted for publication for
c
>
c
∗
with sufficiently small delay time
r
≈
0
. The approach adopted in this paper is the technical weighted energy method used in M. Mei, J.W.-H. So, M.Y. Li, S.S.P. Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579–594, but inspired by Gourley S.A. Gourley, Linear stability of travelling fronts in an age-structured reaction–diffusion population model, Quart. J. Mech. Appl. Math. 58 (2005) 257–268 and based on the property of the critical wavefronts, the weight function is carefully selected and it plays a key role in proving the stability for any
c
>
c
∗
and for an arbitrary time-delay
r
>
0
.
In this paper, the formation of singularities for the nonlocal Whitham-type equations is studied. It is shown that if the lowest slope of flows can be controlled by its highest value with the bounded ...Whitham-type integral kernel initially, then the finite-time blow-up will occur in the form of wave-breaking. This refined wave-breaking result is established by analyzing the monotonicity and continuity properties of a new system of the Riccati-type differential inequalities involving the extremal slopes of flows. Our theory is illustrated via the Whitham equation, Camassa–Holm equation, Degasperis–Procesi equation, and their μ-versions as well as hyperelastic rod equation.
We consider a class of 1D NLS perturbed with a steplike potential. We prove that the nonlinear solutions satisfy the double scattering channels in the energy space. The proof is based on ...concentration-compactness/rigidity method. We prove moreover that in dimension higher than one, classical scattering holds if the potential is periodic in all but one dimension and is steplike and repulsive in the remaining one.
We consider the inhomogeneous Neumann initial–boundary value problem for the nonlinear Schrödinger equation, formulated on a half-line. We study traditionally important problems of the theory of ...nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.
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