In this monograph, we review the theory and establish new and general results regarding spreading properties for heterogeneous reaction-diffusion equations:
∂
t
u
−
∑
i
,
j
=
1
N
a
i
,
j
(
t
,
x
)
∂
i
j
u
−
∑
i
=
1
N
q
i
(
t
,
x
)
∂
i
u
=
f
(
t
,
x
,
u
)
.
\begin{equation*} \partial _{t} u - \sum _{i,j=1}^N a_{i,j}(t,x)\partial _{ij}u-\sum _{i=1}^N q_i(t,x)\partial _i u=f(t,x,u). \end{equation*}
These are concerned with the dynamics of the solution starting from initial data with compact support. The nonlinearity
f
f
is of Fisher-KPP type, and admits
0
0
as an unstable steady state and
1
1
as a globally attractive one (or, more generally, admits entire solutions
p
±
(
t
,
x
)
p^\pm (t,x)
, where
p
−
p^-
is unstable and
p
+
p^+
is globally attractive). Here, the coefficients
a
i
,
j
,
q
i
,
f
a_{i,j}, q_i, f
are only assumed to be uniformly elliptic, continuous and bounded in
(
t
,
x
)
(t,x)
. To describe the spreading dynamics, we construct two non-empty star-shaped compact sets
S
_
⊂
S
¯
⊂
R
N
\underline {\mathcal {S}}\subset \overline {\mathcal {S}} \subset \mathbb {R}^N
such that for all compact set
K
⊂
i
n
t
(
S
_
)
K\subset \mathrm {int}(\underline {\mathcal {S}})
(resp. all closed set
F
⊂
R
N
∖
S
¯
F\subset \mathbb {R}^N\backslash \overline {\mathcal {S}}
), one has
lim
t
→
+
∞
sup
x
∈
t
K
|
u
(
t
,
x
)
−
1
|
=
0
\lim _{t\to +\infty } \sup _{x\in tK} |u(t,x)-1| = 0
(resp.
lim
t
→
+
∞
sup
x
∈
t
F
|
u
(
t
,
x
)
|
=
0
\lim _{t\to +\infty } \sup _{x\in tF} |u(t,x)| =0
).
The characterizations of these sets involve two new notions of generalized principal eigenvalues for linear parabolic operators in unbounded domains. In particular, it allows us to show that
S
¯
=
S
_
\overline {\mathcal {S}}=\underline {\mathcal {S}}
and to establish an exact asymptotic speed of propagation in various frameworks. These include: almost periodic, asymptotically almost periodic, uniquely ergodic, slowly varying, radially periodic and random stationary ergodic equations. In dimension
N
N
, if the coefficients converge in radial segments, again we show that
S
¯
=
S
_
\overline {\mathcal {S}}=\underline {\mathcal {S}}
and this set is characterized using some geometric optics minimization problem. Lastly, we construct an explicit example of non-convex expansion sets.